Wing and fuselage structural optimization considering alternative material
Upcoming SlideShare
Loading in...5
×
 

Wing and fuselage structural optimization considering alternative material

on

  • 789 views

a useful book to optimize Aircraft structure

a useful book to optimize Aircraft structure

Statistics

Views

Total Views
789
Views on SlideShare
789
Embed Views
0

Actions

Likes
0
Downloads
59
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

Wing and fuselage structural optimization considering alternative material Wing and fuselage structural optimization considering alternative material Document Transcript

  • Wing and Fuselage Structural Optimization Considering Alternative Material Systems By Jonathan Lusk B.S.A.E., University of Kansas, 2006Submitted to the Department of Aerospace Engineering and the Faculty of theGraduate School of the University of Kansas in partial fulfillment of the requirementsfor the degree of Masters of Engineering. ______________________________ Professor in Charge-Dr. Mark Ewing ______________________________ Dr. Richard Hale ______________________________ Dr. Stan Rolfe ___________________ Date Project Accepted
  • The Thesis Committee for Jonathan Lusk Certifies that this is the approved Version of the following thesis:Wing and Fuselage Optimization Considering Alternative Material Systems Committee: ______________________________ Professor in Charge-Dr. Mark Ewing ______________________________ Dr. Richard Hale ______________________________ Dr. Stan Rolfe ___________________ Date defended i
  • Abstract This study is a sensitivity analysis to compare weight benefits for a transportaircraft airframe from potential mechanical property enhancements of CFRP (CarbonFiber Reinforced Plastic) Laminate and Aluminum Alloy. The computationalframework is based on a simplified skin-stringer-frame/rib configuration to model thefuselage and the wings of a generic narrow and wide body jet transport. Simple(Strength of Materials) mechanics were used to predict the stresses in the skin andstringers. Strength allowables and panel buckling equations are used in conjunctionwith an iterative optimizer to calculate the structural airframe weight. The baselinematerials include 7075-T6 Aluminum Alloy and a fictitious intermediate moduluscarbon epoxy. For the CFRP material, the optimized weight results show Open Holecompression enhancement produces the most weight benefit. The Fatigue strength isthe most sensitive material property for the baseline Aluminum Alloy structure. Theresults also indicate that the current CFRP laminate minimum gauge limits weightreduction from potential material property enhancements especially on the small jettransport. ii
  • Acknowledgements First of all I would like to thank my academic advisor Dr. Mark Ewing for hisguidance, assistance, advice, and support throughout my project. I would also like tothank Dr. Mark Ewing for teaching me the fundamentals of structural optimizationand introducing me to computational resources to conduct my analysis. I’d like tothank Dr. Abdel Abusafieh and Cytec Engineering Materials for providing fundingfor my research and also providing me their time, information, and resources for meto carryout my analysis. I would also like to acknowledge Dr. Abdel Abusafieh fortaking the time to review my documentation and results and discussing as well asenlightening me on the interest and concerns of the industry. I’d like to thank Dr. Richard Hale for teaching me the mechanics and designof composite materials and making me aware of details that should be considered inthis type of analysis. I’d like to thank Dr. Stan Rolfe for teaching me the factors thatpertain to a complete fatigue analysis and for kindly participating in my examinationboard committee. iii
  • Table of ContentsAbstract........................................................................................................................ iiAcknowledgements...................................................................................................... iiiList of Figure ............................................................................................................. viiList of Tables ............................................................................................................. xiiNomenclature ........................................................................................................... xiv1 Introduction......................................................................................................... 1 1.1 Objective ....................................................................................................... 1 1.2 Prior Work .................................................................................................... 22 Methodology ........................................................................................................ 4 2.1 Geometry....................................................................................................... 4 2.1.1 Wing ..................................................................................................... 4 2.1.2 Fuselage.............................................................................................. 10 2.2 Stiffened Panel Geometry ........................................................................... 11 2.2.1 Wing ................................................................................................... 12 2.2.2 Fuselage.............................................................................................. 15 2.3 Loads........................................................................................................... 18 2.3.1 Wing ................................................................................................... 18 2.3.2 Fuselage.............................................................................................. 27 2.4 Stress Analysis ............................................................................................ 33 2.4.1 Thin Walled Idealized Beam Theory (Ref 2).................................. 33 2.4.2 Hoop and Longitudinal Stress (Ref 5)............................................. 35 iv
  • 2.4.3 Buckling Analysis.............................................................................. 36 2.5 Materials ..................................................................................................... 51 2.5.1 CFRP.................................................................................................. 51 2.5.2 CFRP Lamination Scheme of Aircraft Structure .......................... 52 2.5.3 Aluminum .......................................................................................... 54 2.5.4 Material Enhancement Analysis Cases........................................... 55 2.6 Failure Modes and Criteria ......................................................................... 57 2.6.1 CFRP.................................................................................................. 57 2.6.2 Aluminum Alloy ................................................................................ 60 2.7 Optimization ............................................................................................... 61 2.7.1 Optimized Variables ......................................................................... 61 2.7.2 Fixed Variables.................................................................................. 61 2.7.3 Concept of Minimum Gauge............................................................ 63 2.7.4 Discretization..................................................................................... 63 2.7.5 Optimization...................................................................................... 703 Verification ......................................................................................................... 78 3.1 Regression Analysis.................................................................................... 78 3.1.1 Wing ................................................................................................... 80 3.1.2 Fuselage.............................................................................................. 83 3.2 Finite Element............................................................................................. 86 3.2.1 Wing ................................................................................................... 86 3.2.2 Fuselage.............................................................................................. 89 v
  • 4 Results and Discussion...................................................................................... 925 Conclusions...................................................................................................... 1096 Recommendations ........................................................................................... 110References................................................................................................................ 112A Appendix........................................................................................................... A.1B Appendix............................................................................................................B.1C Appendix............................................................................................................C.1D Appendix........................................................................................................... D.1 vi
  • List of FigureFigure 2.1: Wing Plan form with Kick ......................................................................... 4Figure 2.2: Wing Chord Distribution............................................................................ 6Figure 2.3 Wingbox and Wing Planform...................................................................... 7Figure 2.4: Illustration of Wingbox Thickness Taper (this in not a configurationconcept illustration and there is no rib geometry displayed in this figure)................... 8Figure 2.5: Fuselage Geometry................................................................................... 10Figure 2.6: "I" beam Section Stiffened Panel ............................................................. 12Figure 2.7: Geometry of an "I"-Beam Section Stiffener............................................. 13Figure 2.8: "Hat"-Section Stiffened Panel .................................................................. 15Figure 2.9:Geometry of a Hat Stiffener ...................................................................... 16Figure 2.10: Generic Load Profile for 2.5g Positive Maneuver ................................. 18Figure 2.11: Generic Load Profile for Negative 1.0g Gust......................................... 19Figure 2.12: Trapezoidal Lift Distribution (1g Steady Level Flight) ......................... 20Figure 2.13: Elliptical Lift Distribution (1g Steady Level Flight).............................. 20Figure 2.14: Plot of Trapezoid and Elliptic Chord Distribution ................................ 21Figure 2.15: Lift Distribution...................................................................................... 21Figure 2.16: Wing Structural Weight Distribution ..................................................... 23Figure 2.17: Fuel in Wing Weight Distribution.......................................................... 24Figure 2.18: Distributed Load Summary Profile of Positive 2.5g maneuver (does notinclude engine point load)........................................................................................... 25 vii
  • Figure 2.19: Shear and Moment Diagram (Positive 2.5g maneuver) ......................... 26Figure 2.20: Correction for Wing Sweep Illustration ................................................. 27Figure 2.21: Fuselage Structure Loading Modeled as Cantilever Beam .................... 29Figure 2.22: Shear and Moment Distribution for Fuselage Load Case 1 ( Positive 2.5gManeuver) ................................................................................................................... 31Figure 2.23: Shear and Moment Distribution for Fuselage Load Case 2 (Negative 2.0gHard Landing)............................................................................................................. 32Figure 2.24: Stiffened Panels on the Wing Box Skin ................................................. 37Figure 2.25: Principal Stress Components of Pure Shear Buckling (Ref 2)............... 39Figure 2.26: Bending Buckling Illustration (Ref 7).................................................... 40Figure 2.27: Compression Shear Buckling Interaction (Ref 7) .................................. 42Figure 2.28: One Edge Free Crippling (logarithmic scale) (Ref 4) ........................... 45Figure 2.29: No Edge Free Crippling (logarithmic scale) (Ref 4).............................. 45Figure 2.30: Crippling Curves .................................................................................... 46Figure 2.31: Skin Buckled Stiffened Panel and Effective Width Illustration............. 49Figure 2.32: Lumping Areas on the Wing Box........................................................... 65Figure 2.33:Generic Axial Stress and Skin Thickness Relationship .......................... 73Figure 2.34: Generic Buckling Stability and Axial Stress vs Skin ThicknessRelationship ................................................................................................................ 75Figure 2.35: Optimization Flow Chart (example for fuselage)................................... 77Figure 3.1: Linear Regression Correlation between PDCYL and Calculated Results 81 viii
  • Figure 3.2: :Linear Regression Correlation between Actual Load Bearing Weight andCalculated Results....................................................................................................... 81Figure 3.3: Linear Regression Correlation between Actual Primary Weight andCalculated Results....................................................................................................... 82Figure 3.4: : Linear Regression Correlation between Actual Total Structural Weightand Calculated Results................................................................................................ 82Figure 3.5: Linear Regression Correlation between PDCYL and Calculated ModelResults......................................................................................................................... 83Figure 3.6: Linear Regression Correlation between Actual Load Bearing Weight andCalculated Model Results ........................................................................................... 84Figure 3.7: Linear Regression Correlation between Actual Primary Weight andCalculated Model Results ........................................................................................... 84Figure 3.8: Linear Regression Correlation between Actual Total Structural Weightand Calculated Model Results .................................................................................... 85Figure 3.9: Wing Finite Element Geometry ............................................................... 86Figure 3.10: Axial Stress in Skin ................................................................................ 87Figure 3.11: Top Skin Element ID Numbers .............................................................. 87Figure 3.12: Axial and Shear Stresses in Elements 4213 and 4214 (“F06 file”)........ 88Figure 3.13: Axial and Shear Stress from Model ....................................................... 88Figure 3.14: Axial Stress............................................................................................. 89Figure 3.15: Axial Stress from Model ........................................................................ 90Figure 3.16: Shear Stress ............................................................................................ 91 ix
  • Figure 3.17: Shear Stress Calculated in Model........................................................... 91Figure 4.1: Percent Load Bearing Structural Weight Benifit of CFRP over Aluminumfor a Range of Potential Fatigue Performance........................................................... 96Figure 4.2: Skin Thickness Design Stress Relationship ............................................. 98Figure 4.3: Failure Mode Distribution on Wing Structure for Wide and Narrow body(CFRP Baseline and all cases except for narrow body Improved Composite 1)........ 99Figure 4.4: Failure Mode Distribution for Narrow Body Wing (CFRP ImprovedComposite 1)............................................................................................................. 100Figure 4.5: Failure Mode Distribution on Wide Body Fuselage (CFRP) ................. 101Figure 4.6: Failure Mode Distribution on Narrow Body Fuselage (CFRP) ............. 102Figure 4.7: Failure Mode Distribution for Wide Body and Narrow Body Wing(Aluminum)............................................................................................................... 105Figure 4.8: Failure Mode Distribution for Wide Body Fuselage (Aluminum)......... 107Figure 4.9: : Failure Mode Distribution for Narrow Body Fuselage (Aluminum) ... 108Figure A.1: Medium Body Jet Transport.................................................................. A.1Figure A.2: V-N Diagram Specifications for Military Airplanes (Ref 6)................. A.2Figure B.1:WB Wing Running Loads .......................................................................B.5Figure B.2: NB Wing Running Loads .......................................................................B.5Figure B.3: Crippling of CFRP Laminate "I"-Section Beam With Ex=14.5Msi.......B.6Figure B.4: Euler Column Buckling of CFRP Laminate Ex=14.5Msi .....................B.7Figure B.5: Radius of Gyration for Euler Buckling Calculation with Contribution ofEffective Width..........................................................................................................B.8 x
  • Figure C.1: 7075 Aluminum Alloy Mechanical Properties (Ref 4)...........................C.1Figure C.2: "K" Values for Compression and Shear Panel Buckling (Ref 7) ...........C.2Figure C.3: Stiffness Correction ................................................................................C.3Figure D.1: Illustration of Fuselage Modeled as Idealized Tube.............................. D.1Figure D.2: Optimized Thicknesses for Fuselage Section 46................................... D.1Figure D.3: Lateral Cut of Fuselage as a Idealized Tube ......................................... D.3Figure D.4: Center of Mass (in)................................................................................ D.4Figure D.5: Second Moment of Area (in4)............................................................... D.5Figure D.6: Internal Loads (shear(lbs), moment(lbs*in))......................................... D.6Figure D.7: Maximum and Minimum Principle Stress in Crown (psi) .................... D.7Figure D.8: Minimum Principle Stress in Belly (psi) ............................................... D.8Figure D.9: Max In-plane Shear Stress (psi) .......................................................... D.10Figure D.10: Idealized Stringer Section.................................................................. D.11Figure D.11: Crippling Stress of Belly Stringer ..................................................... D.13Figure D.12: Geometry of Hat Stiffener ................................................................. D.13Figure D.13: Hat Stiffener Including Effective Width ........................................... D.16Figure D.14: Belly Stringer Buckling Strength ...................................................... D.17Figure D.15: Output for WB CFRP Baseline ......................................................... D.19 xi
  • List of TablesTable 2.1: Wing Model Geometry Values.................................................................... 9Table 2.2: Fuselage Model Geometry Values............................................................. 11Table 2.3: Wing Stiffened Panel Geometry Values (CFRP) ...................................... 12Table 2.4: Wing Striffened Panel Geometry Values (Aluminum) ............................. 12Table 2.5: Wing Stringer Geometry (CFRP) .............................................................. 14Table 2.6: Wing Stringer Geometry (Aluminum)....................................................... 14Table 2.7: Fuselage Stiffened Panel Geometry Values (CFRP) ................................. 15Table 2.8: Fuselage Stiffened Panel Geometry Values (Aluminum) ......................... 15Table 2.9:Fuselage Stringer Geometry (CFRP).......................................................... 17Table 2.10: Fuselage Stringer Geometry (Aluminum) ............................................... 17Table 2.11: Fuel Weight Breakdown .......................................................................... 24Table 2.12: CFRP Lamina Properties ......................................................................... 51Table 2.13: CFRP Laminate Properties ...................................................................... 52Table 2.14: 7075-T6 Aluminum Alloy Properties...................................................... 55Table 3.1: Wing Regression Analysis Data ................................................................ 80Table 3.2: Fuselage Regression Analysis Data........................................................... 83Table 4.1: CFRP Baseline and Enhanced Material Weights (% Structural WeightReduction)................................................................................................................... 93Table 4.2: Aluminum Baseline and Enhanced Material Weights (% Structural WeightReduction)................................................................................................................... 94 xii
  • Table 4.3: Stiffened and Un-stiffened Structural Weights.......................................... 95 xiii
  • Nomenclature English Symbol Description Units [A] Linear Inequality Constraint Matrix ~ [C] Non-Linear Constraint Matrix ~ [D] Laminate Flexure Stiffness Matrix in-lb [LB] Optimization Lower Bound ~ [UB] Optimization Upper Bound ~ {b} Non-linear Inequality Limit ~ a Stiffened Panel Length in. A Cross Section Area in2 ai Discrete Lumped Area in2 b Wing Span ft bc Stiffener Cap Panel Width in bf Stiffener Flange Width in bw Stiffener Web Width in c Wing Chord ft D Fuselage Diameter ft Dmn Flexural Stiffness Matrix Term in-lb E Elastic Modulus psi F Stress (Military Handbook Notation) psi xiv
  • f Optimization Cost Function in2g Inequality Constraint ~I Second Moment of Area in4 Distance to Wing Kick (in percent% wing halfk span) ~l Fuselage Length ftL Length of Longitudinal Stiffener in.L Distributed Lift lb/inM Internal Moment Load lb-inn Load Factor ~p Net Distributed Load lb/inq Shear Flow lb/inQ First Moment of Area in3R Critical Buckling Stress Ratio ~S Wing Planform Surface Area ft2T Gross Engine Thrust lbt Thickness inV Internal Shear Load lbW Weight lbw Thin Plate Deflection in.x Structure Axial Coordinate fty Structure Lateral Coordinate in. xv
  • Acronyms Description AR Wing Aspect Ratio ~ M.S. Margin of Safety ~ OHC Open Hole Compression Strength psi OHT Open Hole Tension Strength psi S.F. Safety Factor ~ W.S. Wing Station ft PDCYL NASAs Structural Weight EstimateGreek Symbol Description Δσ Change in Stress Allowable psi Δt Change in Thickness in. λ Taper Ratio ~ Λ Sweep Angle Degrees ν Poissons Ratio ~ θ Stiffener Web Inclination Angle Degrees ρ Radius of Gyration in. σ Axial or Transverse Stress psi τ Shear Stress psi xvi
  • Subscript Description 0 Initial Value 1 Maximum Principal Direction/Fiber Direction 2 Minimum Principal Direction/Transverse Direction Laminate Flexural Stiffness Response to In-Plane Bending 11 Strain Laminate Flexural Stiffness Response to Transverse 12 Bending Strain Laminate Transverse Flexural Response to Transverse 22 Bending Strain Laminate Torsional Flexure Response to In-Plane Torsional 66 Strain 1,2i Principal Stress of a Discrete Lumped Area 12su Lamina Ultimate Shear Strength 1cu Lamina Ultimate Compression Strength in Fiber Direction 1tu Lamina Ultimate Tensile Strength in Fiber Direction Lamina Ultimate Compression Strength Transverse to Fiber 2cu Direction Lamina Ultimate Tensile Strength Transverse to Fiber 2tu Direction All Allowable Stress b Bending Stress xvii
  • c Compression Stress cc Crippling Strength CL Wing Center Line cr Critical Buckling Strength cr_stiff Critical Bucklng Strength of Stiffener cu Ultimate Compression Strength Design Design Stress e Effective Length engine/pylon Engine and Pylon Weight exp Exposed Wing Area Fracture Margin of Safety/Flange Width/ Forward Fuselage f Location/Fiber Volume fi Final Thickness of Discrete Structural Area Fract_sk_Bottom Critical Fracture of Bottom Skin Fract_sk_Top Critical Fracture of Top Skin Fract_st_Bottom Critical Fracture of Bottom Stiffener Fract_st_Top Critical Fracture of Top StiffenerFracture_Allowable Fracture Strength fuselage Fuselage Cross Secitonal Area G Center of Area GTO Gross Takeoff Weight h High Relative Stress Range xviii
  • i Discrete Area, Thickness or Stress Notation xix
  • Subscript Description kick Wing Thickness/Depth at Wing Kick l Low Relative Stress Range L Longitudinal Stress Ratio LE Leading Edge Sweep/ Swept Span Limit Stress at Limit Load Linear Linear Inequality Constraint MGTO Maximum Gross Takeoff WeightNon-Linear Non-Linear Inequality Constraint OHC Open Hole Compression Strength OHT Open Hole Tension Strength oi Initial Thickness of Discrete Lumped Structure p Panel Geometry r Root Chord/Thickness r_wb Root Chord of Wing Box ref Reference s Shear Stress Ratio skin Fuselage Cross Section Skin AreaSpar_Caps Wing Cross Section Spar Caps AreaSpar_Webs Wing Cross Section Spar Web Area stiff Stress in Stiffener xx
  • stringers Wing/Fuselage Cross Section Stringer Area su Shear Ultimate t Tip Chord/Thickness t_wb Tip Chord of Wing Box tu Tensile Ultimate wing Wing Cross Section Area x In-Plane Axial Mechanical Properties/Stress In-Plane Axial Mechanical Properties/Stress of Discrete xi Structure xy In-Plane Shear Mechanical Properties/Stress In-Plane Shear Mechanical Properties/Stress of Discrete xyi Structure y Transverse Axial Mechanical Properties/Stress Lateral First/Second Moment of Area (Referenced Text z Notation)z_LC1 Vertical Load Factor for Load Case 1z_LC2 Vertical Load Factor for Load Case 2 Lateral First/Second Moment of Area (Referenced Text zz Notation) xxi
  • 1 Introduction1.1 Objective The main objective of this study is to determine if potential mechanicalproperty enhancements of CFRP material would lend themselves to the application onsmaller transport aircraft, referred to as the narrow body aircraft. This is determinedby identifying and comparing critical failure modes of CFRP and Aluminum Alloyson medium and small commercial jet transport. It is currently known that CFRPmaterials show benefits over Aluminum Alloys for current medium jet transportaircraft, referred to as the wide body aircraft. This study is a weight sensitivityanalysis only and does not take into account factors such as Acquisition and LifeCycle cost and is not intended to compare Aluminum Alloy to CFRP Laminatematerials. The general questions that this analysis addresses are:How does the weight performance benefits from the application of CFRP andAluminum Alloy on a medium transport aircraft compare with that on a smalltransport aircraft?What CFRP material enhancements offer the most weight benefit on the wideand narrow body aircraft and what are the critical failure modes? 1
  • What Aluminum Alloy material enhancements offer the most weight benefit onthe wide and narrow body aircraft and what are the critical failure modes? This study presents a method of estimating the wing box and fuselagegeometry from fundamental preliminary design parameters. Also presented is ananalytical method using optimization to evaluate the weight benefits of altering thematerial properties of the structure. This optimization model will be useful instudying the effects and limitations of enhancing aspect of a unidirectional materialon the wing and fuselage structure. This method determines the primary load bearingstructural mass by sizing the skin and the stringers, but not the fuselage frames orwing ribs. The only input parameters in this analysis are mechanical properties of thematerial. Therefore there is not a configuration trade study included and theconfiguration concept is fixed, though the dimensions of the structural concept areoptimized for the specific baseline material and transport aircraft size.1.2 Prior Work A parallel study was found in a NASA Technical Memorandum Titled:Analytical Fuselage and Wing Weight Estimation of Transport Aircraft. Thedocument presents an methodical procedure in defining the structure geometry, loads,and failure criteria in order to estimate the structural load, similar to the procedurepresented in this document. Unlike this study, the procedure does not include 2
  • compression and shear buckling interaction, sub-structure of different in-planestiffness, and did not study “I”-beam and “Hat”-section stiffened concepts. It doesinclude, unlike this study, curvature of fuselage geometry, deflection analysis, thesizing and weight of fuselage frames and wing ribs, and the study of “Z” –sectionand Sandwich Honeycomb stiffened configurations. The technical memorandum tabulates the actual structural weights of eightAluminum Alloy transport aircrafts and compares the weights with the calculatedstructural weight of their model. This study uses those published weights to verify theoptimized weight calculations for the eight transport aircraft using the optimizedmodel documented in this study. The results are presented in the verification section. 3
  • 2 Methodology2.1 Geometry2.1.1 Wing The wing planform geometry for a transport aircraft is estimated using FigureA.1 in Appendix A. Figure A.1 shows that the wing does not have a straight taper, buta “kick” from the edge of the fuselage to the position where the engine attaches to thewing. The wing planform with the “kick” is illustrated in Figure 2.1. The kick k ,which is given as a fraction of the half span, is located where there is a geometrytransition and the wing taper is discontinuous. D Fuselage Wall (Root) 2 Λ LEc CL cr k ct b 2 Figure 2.1: Wing Plan form with Kick 4
  • Figure 2.1 illustrates the geometry for the wing for the wide body jettransport. The known wing geometry includes the Aspect ratio AR , span b , andleading edge sweep Λ LE . The wing reference surface area S ref is determined from theaspect ratio and span in Equation 1. The span, reference surface area, and the chorddistribution for the wide body, presented in Figure 2.2, is used to calculate the meangeometric chord c . The formula used to calculate mean geometric chord is shown inEquation 2. This chord distribution is determined from Figure A.1 in Appendix A.Since the chord distribution is known the root chord c r and the tip chord ct areknown. Figure 2.1 shows that the leading edge has a constant sweep. The quarterchord sweep Λ c 4 is determined by Equation 3. The exposed wing reference areaS exp is equal to the reference wing area minus the referenced wing area within thefuselage diameter (or fuselage wall) shown in Figure 2.2. ( 1) b2 AR = S ref ( 2) b 2 2 c= ∫c 2 dy S ref 0 5
  • ( 3) 3 tan Λ c = tan Λ LE 4 4 c( y ) cCL cr ct y D k 2 b 2 Figure 2.2: Wing Chord Distribution The wing box structure extends the span of the wing and the width of the wingbox is a fraction of the chord for a specific wing station. The wing box is shownwithin the geometry of the wing planform in Figure 2.3. It is assumed for simplicityof analysis that the center of the wing box is located at the quarter chord of the wingplanform which is also assumed to be the location of the aerodynamic center andcenter of pressure. This implied that the aerodynamic resultant force is located at thecenter of the wing box and that there is no pitching moment on the wing. Theconfigurations for the wide and narrow body transport structures are both two sparconcepts with rib spacing indicated by the stiffened panel geometry length. 6
  • c r _ wbcr c t _ wb ct Figure 2.3 Wingbox and Wing Planform The wing-box thickness (not to be confused with skin thickness) and wing boxwith taper transition at the “kick” is illustrated in Figure 2.4. This is an importantdetail to recognize because there is a large thickness taper from the root to the “kick”and a subtle thickness taper from the “kick” to the tip, which has a large effect on therunning load profile distribution over the wing span. There is a beam that runs behindthe rear landing gear, from the reference wing planform centerline to the “kick” thisbeam is neglected in this analysis because the details needed to size this beam werenot readily available. The geometry parameters of the wing are presented in Table2.1. 7
  • Wingbox to fuselage Intersection Front Spar tr t kick Rear Spar ttFigure 2.4: Illustration of Wingbox Thickness Taper (this in not a configuration concept illustration and there is no rib geometry displayed in this figure) 8
  • Table 2.1: Wing Model Geometry ValuesParameter Parameter Name and Narrrow Body (NB) Wide Body (WB)Designation Description Units Value Value Maximum Gross Takeoff WGTO Weight lbs 123,675 453,000 b Wing Span ft 112 170 D Fuselage Diameter ft 13 19 2 AR Aspect Ratio, AR=b /Sref ~ 9.55 9.55 ΛLE Leading Edge Sweep degrees 40 40 Reference Chord Taper λref Ratio, λref=ct/cCL ~ NA NA Exposed Chord Taper λexp Ratio, λexp=ct/cr ~ 0.1875 0.1875 ⎛t⎞ Thickness to Chord Ratio ⎜ ⎟ ⎝ c ⎠r at Wing Root, (t/c)r= tr/cr ~ 0.10 0.10 Exposed Thickness Ratio, τexp τexp= tt/tr ~ 0.1883 0.1883 Wingbox to Wing Planform Chord Ratio at rr Root, rr=cr_wb/cr ~ 0.41 0.41 Wingbox to Wing Planform Chord Ratio at rt Tip, rt=ct_wb/ct ~ 0.31 0.31 Distance to Kick from Wing Root (as fraction of wing half span), kick kick=k/(b/2) ~ 0.254 0.254 ⎛t⎞ Thickness to Chord Ratio ⎜ ⎟ ⎝ c ⎠k kick, (t/c)k=tkick/ckick ~ 0.10 0.10 9
  • 2.1.2 Fuselage The Fuselage is modeled as an un-tapered cylinder neglecting wing andempennage attachment structure. The maximum diameter of the actual fuselagestructure is used far the diameter of the cylinder. The length of the cylinder extendsfrom the nose to the tail cone. WBD 19ft l Figure 2.5: Fuselage Geometry ( 4)l = 0.67WGTO 0.43 (Ref 2) The fuselage geometry is defined in terms of the fineness ratio l D , and themaximum gross takeoff weight. The length and diameter of the fuselage areillustrated in Figure 2.5. The Gross Takeoff Weight and fuselage slenderness ratio forthe wide and narrow body aircraft is given in Table 2.2. Equation 4 estimates thefuselage length from the gross takeoff weight. 10
  • Table 2.2: Fuselage Model Geometry Values Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value WGTO Gross Takeoff Weight lb 123,675 453,000 l D Fuselage Length to Diameter Ratio (~) 7.8 9.62.2 Stiffened Panel Geometry In order to compare the stability of the stiffened panels at the proper stress levelthe stiffener spacing and geometry is found using an iterative process presented inAppendix B. The “I”-beam section is used to stiffen the wing skin panels and the“Hat”-section stiffener is used on the fuselage. Figure 2.6 shows an “I” beamstiffened panel using a finite element modeling software. Figure 2.8 shows a solidmodel of a hat stiffened panel with the panel width b and panel length a defined. Tables 2.3 and 2.4 give the wing stiffened panel geometry values for CFRP andAluminum Alloy respectively. Tables 2.7 and 2.8 give the stiffened panel geometryfor the CFRP and Aluminum Alloy fuselage. The stiffened panel geometry iscompared for CFRP and Aluminum Alloy for the specific wide or narrow body wingor fuselage structural with the un-stiffened structural weight in the results. The b tstiffener spacing to panel thickness ratio is a function of material stiffness E , stress inthe panel σ x , Poisson’s ratio ν , and boundary conditions. The panel width to skin 11
  • thickness ratio b t should be similar for every like material geometry combination(i.e. The Aluminum Alloy narrow and wide body wing structure).2.2.1 Wing Figure 2.6: "I" beam Section Stiffened Panel Table 2.3: Wing Stiffened Panel Geometry Values (CFRP) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 7.0 10.6 17.0 a Panel Length in 14.0 24 34.0 Table 2.4: Wing Striffened Panel Geometry Values (Aluminum) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 2.3 10.6 5.7 a Panel Length in 4.6 24 11.4 12
  • Figure 2.7 shows the geometry of the “I”-section stiffener. They i displacements are used as inputs in determining the radius of gyration and effectivewidth in the model, there values are not tabulated in this section. Table 2.5 and 2.6give the geometry values of the CFRP and Aluminum Alloy stiffeners for the wingrespectively. The stiffener is assumed to be bonded to the wing skin. The “I” beamstiffener configuration is chosen because it is what is currently being used in industryon a medium transport aircraft with CFRP as its primary load bearing material. bc y ct bw tw yw y tc y cb ts Figure 2.7: Geometry of an "I"-Beam Section Stiffener 13
  • Table 2.5: Wing Stringer Geometry (CFRP) Parameter Parameter Name Narrow Body (NB) Wide Body (WB)Designation and Description Units Value Value Stiffener I-Section bw Web Width in. 1.00 2.00 Stiffener I-Section bf Flange Width in. 0.55 1.30 Table 2.6: Wing Stringer Geometry (Aluminum) Parameter Parameter Name Narrow Body (NB) Wide Body (WB)Designation and Description Units Value Value Stiffener I-Section bw Web Width in. 0.40 1.00 Stiffener I-Section bf Flange Width in. 0.25 0.65 14
  • 2.2.2 Fuselage a b Figure 2.8: "Hat"-Section Stiffened Panel Table 2.7: Fuselage Stiffened Panel Geometry Values (CFRP) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 6.3 10.6 11.0 a Panel Length in 9.4 24 16.4 Table 2.8: Fuselage Stiffened Panel Geometry Values (Aluminum) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 2.5 10.6 4.4 a Panel Length in 3.8 24 6.5 15
  • Figure 2.8 shows the geometry of the “Hat”-section stiffened panel. The websof the stiffener are actually at an angle like what is shown in Figure 2.9. The angle ofthe hat stiffener web is used when determining the radius of gyration of the stiffener.The flange of the stiffener, with width b f in Figure 2.9, is attached to the skin of thefuselage. All flanges are assumed to be bonded to the fuselage skin. Like the “I”-section stiffener the y i lateral locations of the legs of the stiffener are inputs in themodel to compute the radius of gyration and are functions of the stiffener dimensionsand the wing or fuselage skin thickness (i.e. f (bi , t ) ). The geometry of the “Hat”-section stiffener is given in Table 2.9 and 2.10 for the CFRP and Aluminum Alloymaterial respectively. The “Hat”-section stiffener is chosen because it is currentlybeing used in industry on a medium transport aircraft with CFRP as its primary loadbearing material. The stiffened panel geometry given in this section is a fixed input inthe model and is not allowed to vary since the scope of this study is reduced tobenefits of the material mechanical properties only. The stiffened panel geometry isoptimized prior to the sensitivity analysis. bc θ bw y yc yw yf bf *yi is lateral location of bi Figure 2.9:Geometry of a Hat Stiffener 16
  • Table 2.9:Fuselage Stringer Geometry (CFRP) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB)Designation Description Units Value Value Stiffener Hat-Section bw Web Width in. 0.66 1.00 Stiffener Hat-Section bf Flange Width in. 0.50 0.50 Stiffener Hat-Section bc Cap Width in. 1.00 1.50 Stiffener Hat-Section θ Web Angle Degrees 43 43 Table 2.10: Fuselage Stringer Geometry (Aluminum) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB)Designation Description Units Value Value Stiffener Hat-Section bw Web Width in. 0.35 0.70 Stiffener Hat-Section bf Flange Width in. 0.30 0.45 Stiffener Hat-Section bc Cap Width in. 0.70 1.40 Stiffener Hat-Section θ Web Angle Degrees 43 43 17
  • 2.3 Loads2.3.1 Wing The critical load cases being analyzed is a positive 2.5g maneuver and anegative 1.0g gust at limit load that is a standard specification for Jet TransportAircraft (Ref 7). Figure 2.10 and 2.11 illustrates the difference between the two loadcases. Figure A.2, in Appendix A, shows the V-N diagram for transport militaryaircraft. The load case for the medium transport military aircraft is the same for thecommercial medium transport. The medium transport loading capability is marked inFigure A.2. All load cases are symmetric about the center axis of the fuselage. Thereis no torsion, drag, or dynamic load cases applied to the wing because the detailsrequired for these load cases is out of the scope of this sensitivity analysis. b 2 ∫ b L ( x ) dx = 2 . 5 W MGT O − 2 L(x) 2.5WMGTO b Figure 2.10: Generic Load Profile for 2.5g Positive Maneuver 18
  • b 2 ∫ L ( x)dx = −W b MGTO − 2 b W TO Figure 2.11: Generic Load Profile for Negative 1.0g Gust2.3.1.1 Lift Profile Using Shrenk’s Approximation (Ref 3) it is required to find the trapezoidalchord distribution of the wing, shown in Figure 2.12, and the theoretical ellipticalchord distribution of the wing illustrated in Figure 2.13. According to Shrenk’sapproximation the lift distribution is proportional to the average of the Trapezoidaland the elliptic chord distribution. Figure 2.14 plots the comparison between the twodistributions. Figure 2.15 shows the resultant lift distribution of the two geometricallyaveraged. Figure 2.15 gives a linear approximation of the lift distribution which canbe used as a short hand method to assure that the distribution sums to the grosstakeoff weight; this calculation is shown in Equation 5 (453,000 lb is the grosstakeoff weight of the wide body aircraft). Half the gross takeoff weight would beexperienced by the half span of the wing in steady level 1.0g flight. 19
  • Lift per unit length (lb/ft) S=Area Chord(y) W GTO W 37.77 S1 167.16 2..12 GTO 1 2.12 b 88 b S2 557.43 S3 424.82 S4 363.91 S1 Swing 1513.32 W GT ) W 1.1.33 GTO 17 GTO 32 23.61 bb S3 S2 W W 0.40 GTO 0..39 GTO 0 35 GTO 7.08 b b S4 0.055b 23.61 b 0.127 b 182 00.5b y Wing Stationb Wing Station (ft) .500 75.5 ft Figure 2.12: Trapezoidal Lift Distribution (1g Steady Level Flight) Lift per unit length W W 1127 GTO . .13 GTO bb Wing Station (ft) 0.055b 0.500b Figure 2.13: Elliptical Lift Distribution (1g Steady Level Flight) 20
  • 40.0 35.0 30.0Chord Distribution (ft) 25.0 Trapezoid 20.0 Elliptic 15.0 10.0 5.0 0.0 0.0 20.0 40.0 60.0 80.0 Wing Station (ft) Figure 2.14: Plot of Trapezoid and Elliptic Chord Distribution Lift Distribution y = -39.513x + 4406.9 2 R = 0.9966 5000.0 4500.0 4000.0 Lift Distribution (lb/ft) 3500.0 3000.0 Lift 2500.0 Linear (Lift) 2000.0 1500.0 1000.0 500.0 0.0 0.0 20.0 40.0 60.0 80.0 W.S. (ft) Figure 2.15: Lift Distribution 21
  • ( 5) ⎡⎛ lb ⎞ ⎤ ⎢ ⎜ 4500 − 1500 ⎟(75 ft ) lb ⎜ ⎟ ⎥ ⎢⎝ ft ft ⎠ ⎛ lb ⎞Total _ Load = 2 + ⎜1500 ⎟(75 ft )⎥ = 450,000 ≈ 453,000lb ⎜ ⎢ 2 ⎝ ft ⎟ ⎠ ⎥ ⎢ ⎥ ⎢ ⎣ ⎥ ⎦2.3.1.2 Inertial Relief2.3.1.2.1 Wing Weight The wing total structural weight is modeled with wing trapezoidal chorddistribution, shown in Figure 2.16, which is similar to the trapezoidal approximationof the lift distribution. The weight of the wing is subtracted from the lift profile. Thisis consistent with the load case because when the wing is experiencing a positive 2.5gmaneuver the wing weight is resisting the motion with the same load factor. The wingtotal structural weight is estimated to be 12% of the Gross Takeoff Weight. The wingstructural weight distribution is a fixed function of the aircraft gross takeoff weight ofthe aircraft regardless of the material applied to the structure this is done becausethere is a large amount of the structural mass in the wing that cannot be accounted forin this sensitivity analysis. 22
  • 0.055b 0.182b 0.500b Wing Station (ft) W W −00042 GTO . .05 GTO b b W W − 0..14 GTO 016 GTO W wing=50% W structural W structural 108600 lb b b W wing 54300 lb W W − 0..25 GTO 0 23 GTO b Wing Weight Distribution (lb/ft) Figure 2.16: Wing Structural Weight Distribution2.3.1.2.2 Engine and Pylon Weight The engine weight load is applied at the location of the wing kick (25.4% ofthe wing half-span outboard of the wing-fuselage intersection). The weight of theengine and the pylon is applied as a discrete point load on the wing. There are twoengines, one on each wing, for both the wide and narrow body aircraft. Equation 6estimates the engine pylon weight using the gross takeoff weight and the maximumthrust to weight ratio. ( 6) 1.1 ⎛ 1 T ⎞ WEngine / Pylon = 0.064⎜ ⎜ # _ of _ Engines W WGTO ⎟ ⎟ ⎝ ⎠ (Ref 3) 23
  • 2.3.1.2.3 Fuel Weight It is assumed that the portion of the total fuel in the wing, specified in Table2.12, is 80%. The fuel distribution, shown in Figure 2.17, is assumed to taper linearlyto zero within the span of one wing. The fuel weight, along with the engine and wingstructural weight, is subtracted from the positive lift distribution for the resultantloading on the wing structure. Table 2.11: Fuel Weight Breakdown Fuel % Weightfuel 35%W TO %Weight fuel in wing 80%W fuel %bspan occupied by fuel 100%bspan 0.055b 0.500b Wing Station (ft) W - 0.315 WGTO −0.63 GTO b b Fuel Weight Distribution (lb/ft) Figure 2.17: Fuel in Wing Weight Distribution 24
  • 2.3.1.3 Load Summary The lift distribution for the positive 2.5g load case is given in Figure 2.18.The negative 1.0g load case would take the same profile but in the negative liftdirection with 40% of the magnitude. Both load cases on the wing structure are quasi-static and symmetric. 1.000 0.900 0.800 0.700 (lb/ft) 0.600 0.500 Load Distribution Location of Engine 0.400 0.300 0.200 0.100 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 Wing Station× b (ft) Figure 2.18: Distributed Load Summary Profile of Positive 2.5g maneuver (does not include engine point load) 25
  • 2.3.1.4 Internal Loading Engine Point Load Distributed Load 0.3 0.25 (lb) 0.2 Shear × WGTO Shear 0.15 0.1 0.05 0 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 Wing Station 6.0 5.0 (lbs-ft) 4.0 Moment × WGTO b 3.0 2.0 1.0 0.0 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 Wing Station × b (ft) Figure 2.19: Shear and Moment Diagram (Positive 2.5g maneuver) 26
  • 2.3.1.5 Load Distribution Correction for Sweep The load distribution p(x ) is reduced to its component perpendicular to thequarter chord swept span, to account for the effects of a wing sweep.. This will stretchthe shear distribution over the swept span while maintaining the same shear loading atthe root. The moment at the root will increase. The calculation for quarter chordsweep is given in Equation 3. Λ LE Quarter Chord Span b c 4 Λ c 4 C ⎛ ⎞ 4 p((x ) cos⎜ Λ c ⎟ C x) ⎜ ⎟ ⎝ 4 ⎠ c 4 b Figure 2.20: Correction for Wing Sweep Illustration2.3.2 Fuselage The fuselage structure is sized using two load cases: a positive 2.5g maneuverand a negative 2.0g hard landing. Static load cases are used, for this sizing analysis, 27
  • which exclude external aerodynamic pressure forces from drag or bending momentsfrom thrust lines of the engines. The hard landing load case takes into account thenose down pitch rate of the fuselage. This model only takes into account symmetrical load conditions, thereforethere is no torsion induced on the fuselage structure. Two load cases are appliedindividually. The worst case scenario for the crown and belly of any fuselage sectionis taken for a given failure mode. The model does account for pressurization, butneglects the effects of pressurization on fuselage sections in compression and sectionsthat are failure critical from load case 2. This is because it is assumed that thefuselage is not fully pressurized in the negative 2.0g hard landing load case.2.3.2.1 Loading Profile The loading profile for the fuselage structure includes the fuselage structuralweight, the airplane payload weight, the front landing gear weight, and theempennage structural weight. It is assumed that the rear landing gear is mounted inthe wing fuselage attachment and therefore the load is taken by structural loadingpoints that are not sized in this model. The loading profile is illustrated in Figure 2.21. 28
  • Fixed at Airplane Front Landing Gear Wing Load Empennage Load x1 x2 xf xt Fuselage Structure Distributed Load Passenger/Cargo Distributed Load Figure 2.21: Fuselage Structure Loading Modeled as Cantilever Beam The fuselage is modeled as two cantilever beams facing outwards, as shown inFigure 2.21, from the wing box quarter chord point. Modeling the fuselage as twocantilever beam is convenient because modeling it as one beam makes it anindeterminate problem. Each cantilever beam has its own coordinate referencesystem. x f is the fuselage station coordinate starting at the nose of the fuselage andending at the fuselage quarter chord. xt is the coordinate originating from the tail ofthe airplane and ending at the wing box quarter chord point.2.3.2.2 Internal Loading For the loading given in Figure 2.21 the bottom of the fuselage is incompression which corresponds to a negative bending moment. With the signconvention in Equation 7 and 8, the coordinate system using x f and x t , a maximumpositive shear, and minimum negative bending moment takes place at the wing box 29
  • quarter chord. The internal load distribution for load case 1 and 2 are given in Figures2.22 and 2.23 respectively. ( 7) V y ( x) = ∫ − p( x)dx ( 8) ∂M x = −Vy ∂x 30
  • Tension Compression 41 43 44 46 47 48 541in. 276in. 336in 396in 273in 379in. WB 0.2 0.18 0.16 Shear × GTO (lb) Shear × WWGTO (lb) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fuselage Station × b (ft) Fuselage Station ×b (ft) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 (lbs-ft) × 10 × WGTO WGTO b (lbs-ft) -1 -2 × 10 −2 × b -3 Moment −2oment -4 M -5 -6 Fuselage × b × b (ft) (ft) Fuselage Station StationFigure 2.22: Shear and Moment Distribution for Fuselage Load Case 1 ( Positive 2.5g Maneuver) 31
  • Tension Compression 41 43 44 46 48 47 541in. 276in. 336in 396in 273in 379in. WB 0.7 0.6 0.5 0.4 (lb) 0.3 × WGTO 0.2 Shea 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 -0.2 -0.3 Fuselage Station ×Station Fuselage b (ft) × b (ft) 6 4 2 Moment × 10 −2 × WGTO b (lb-ft) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 -4 -6 -8 -10 -12 -14 Fuselage Station × b (ft) Fuselage Station ×b(ft)Figure 2.23: Shear and Moment Distribution for Fuselage Load Case 2 (Negative 2.0g Hard Landing) 32
  • The shear and moment distribution profiles are the same for the narrow andwide body, though they have different magnitudes and length of distributions. Themaneuver load case has a maximum negative moment at the intersection of the rearfuselage and the wing box. The hard landing condition has a maximum negativemoment at the rear landing gear and a maximum positive moment on the forwardfuselage at the mid point between the front and rear landing gear.2.3.2.3 PressurizationThe fuselage skin has a tensile stress from an ultimate pressurization load of 18 psi,applied to all fuselage cabin structure. The pressurization load is the same betweenthe narrow and wide body aircraft, and also the same for the analysis of the CFRP andAluminum Alloy materials.2.4 Stress Analysis2.4.1 Thin Walled Idealized Beam Theory (Ref 2)The axial stress formula is given in Equation 9. M z is the moment about the lateralaxis. y i is the discrete location of stress. 33
  • ( 9) − M z ( yi − yG ) σx = I ZZThe Shear Flow and Shear Stress formula is given in Equation 10 and 11. ( 10) V y Qz Δq = I zz , ( 11) qi τ xy = tiQ z and I z are the first and second moment of area respectively, and are given inEquation 12. The first and second moment of inertia using discrete areas is given inEquation 13. ( 12) Q z = ∫∫ ydA A I z = ∫∫ y 2 dA A ( 13) n Q z = ∑ Ai ( y i − y G ) i =1 n I z = ∑ Ai ( y i − y G ) 2 i =1 34
  • y G is the lateral neutral axis (or y-component of the mass center) and its formula isgiven in Equation 14. The discretized calculation of the y-component of the centroidis also in Equation 14. ( 14) n 1 ∑Ay i i yG = ∫∫ ydA = i =1 n ∑A A A i i =12.4.2 Hoop and Longitudinal Stress (Ref 5) The hoop stress is the circumferential stress in a pressure vessel wall. Thelongitudinal stress is parallel to the axis of symmetry in the pressure vessel wall. p isthe pressure in psi. r is the radius of the pressure vessel. t is the thickness of thepressure vessel wall. ( 15) pr σ Hoop = t pr σ Longitudinal = 2t 35
  • 2.4.3 Buckling Analysis2.4.3.1 Compression Buckling The buckling strength of a panel depends on thickness, width, stiffness, aspectratio, Poisson’s effects and the lay-up or bending stiffness matrix of the laminate.Figure 2.24 shows the geometry and loads on individual panels of width b and lengtha that are used to determine buckling stability. Since the buckling strength forlaminates depends on thickness and stacking sequence the flexural stiffnesscoefficients are used, referred to as the [D ] matrix. The [D ] matrix terms are used tocalculate the buckling strength of a composite laminate in Equation 16. N x is the critical running load in the laminate panel expressed in (lb/in.). m isthe number of half sine waves formed by the buckled material. The value of m ischosen that gives the minimum buckling strength for a given aspect ratio. ( 16) π2 ⎡ ⎤ 2 2 2 ⎛b⎞ N x = 2 ⎢ D11 m ⎜ ⎟ + 2 2 (D12 + 2 D66 ) + D22 ⎛ a ⎞ ⎛ 1 ⎞ ⎥ ⎜ ⎟ ⎜ ⎟ b ⎢⎣ ⎝a⎠ ⎝b⎠ ⎝m⎠ ⎥ ⎦ 36
  • Panel for Buckling Analysis Aspect Ratio = a/b a b a b Compression Stringers Shear ribs Figure 2.24: Stiffened Panels on the Wing Box Skin Buckling of aluminum depends on thickness, width, stiffness, poisons ratioand aspect ratio. The equations to calculate buckling strength are for isotropicmaterials. The isotropic buckling equations are used to calculate the buckling strengthof an Aluminum Alloy panel as shown in Equation 17, where E c is the ElasticModulus of the material in compression. The plot illustrating the K values for thecompression and shear buckling are in Appendix C, Figure C.2. ( 17) π 2 Ec ⎛ t ⎞ 2 2 ⎛t⎞ ⎛a⎞ σ cr = C c 2 ⎜ ⎟ = K c E c ⎜ ⎟ where K c = f ⎜ ⎟ 12(1 − ν ) ⎝ b ⎠ ⎝b⎠ ⎝b⎠ 37
  • 2.4.3.2 Shear Buckling The shear buckling strength formula in Equation 18 is found in Reference 8for a composite laminate. It is also a function of the flexure coefficients. Thisequation is derived assuming the long plate assumption which makes it a conservativeshear buckling strength calculation for the geometry of the panels in this structure. Kis a parameter that accounts for bend-twist coupling in the stability of a laminate. ( 18) D12 + 2 D66 K= D11 D22 D11 D22 (8.125 + 5.045K ) when K ≤ 1 4 N xy = 4 3 b2 ⎛ 1.46 ⎞ D22 (D12 + 2 D66 )⎜11.71 + 2 ⎟ when K < 1 4 N xy = 4 b2 ⎝ K ⎠ In Equation 19 the shear buckling strength τ cr , for the isotropic AluminumAlloy, is a function of axial Elastic Modulus E . Equation 19 uses the ElasticModulus rather than the shear modulus, because pure shear produces equalcompressive and tensile principal stresses on the diagonal plane with respect to theedge of the plate. The concept of pure shear as diagonal principal stresses is displayedin Figure 2.25 (Ref 2). 38
  • ( 19) π 2 Ec ⎛ t ⎞ 2 2 ⎛t⎞ ⎛a⎞ τ cr = Cs 2 ⎜ ⎟ = K s E c ⎜ ⎟ where K s = f ⎜ ⎟ 12(1 − ν ) ⎝ b ⎠ ⎝b⎠ ⎝b⎠ τ τ σ 45 o τ σ τ τ τ σ τ σ τ Figure 2.25: Principal Stress Components of Pure Shear Buckling (Ref 2)2.4.3.3 Bending Buckling The bending buckling strength formula, Equation 20, for a composite laminateis found in Reference 8. Like the shear buckling equation for composite laminate,Equation 20 assumes the long plate assumption. ( 20) Nx = π2 b2 [13.9 ] D11 D22 + 11.1(D12 + 2 D66 ) 39
  • The bending buckling allowable, used for the Aluminum Alloy, is found usingEquation 21 (Ref 7). The buckling coefficient k b is found from Figure C5.15 fromReference 7. The Elastic Modulus for Compression E = E c is used for AluminumAlloy in Equation 21. Bending buckling is illustrated, in Figure 2.26, where w is thedeflection of the web. Buckling, in Figure 2.26, is positive for “out of the page” andnegative for “in the page”. ( 21) π 2 Ec ⎛ t ⎞ 2 σ b = kb ⎜ ⎟ 12(1 − ν 2 ) ⎝ b ⎠ w=deflection 2 b 3 σb w+ w- w+ w- w+ w- w+ w- b w=0 0 0 0 a Figure 2.26: Bending Buckling Illustration (Ref 7) 40
  • 2.4.3.4 Compression-Shear Interaction To determine the compression and shear buckling interaction the followingratios are defined in Equation 22. The margin of safety for this buckling interaction isdefined in Equation 23 and is displayed in Figure 2.27 for a margin of safety of zero.The critical compression ratio RL for axial stress is negative if the axial stress is intension and positive if in compression. If RL is negative it helps prohibit shearbuckling, therefore increasing the buckling stability in Equation 23. ( 22) σx RL = σ Cr τ RS = x τ Cr (Note: RL is negative for tension) ( 23) 2 M .S . = −1 R L + R L + 4 RS 2 2 41
  • 1.6 1.4 Shear & Tension 1.2 M.S.=0 1 RL+RS2=1 Pure Shear Buckling Compression & Shear Rs 0.8 0.6 Pure Compression Buckling 0.4 0.2 Tension Compression Tensio 0 -1.0 -0.5 0.0 0.5 1.0 RL Figure 2.27: Compression Shear Buckling Interaction (Ref 7)2.4.3.5 Shear-Bending Interaction The shear-bending buckling interaction formula uses the ratios presented inEquation 24 and 25. The ratios include: the shear stress τ xy and shear bucklingstrength τ cr ratio. The bending axial stress on a web σ x _ b , shown in Figure 2.25, andthe bending strength σ cr _ b ratio. The ratios in Equation 24 are used in Equation 25 todetermine shear-bending buckling failure. 42
  • ( 24) τ xy Rs = τ cr σ x_b Rb = σ cr _ bending ( 25) 1 M .S . = −1 Rs + Rb 2 22.4.3.6 Longitudinal Crippling of Stringers2.4.3.6.1 CFRP A method for determining the crippling allowables for a long slender plate istaken from Reference 4. “Tests conducted by Lockheed and McDonnell Douglasunder their Independent Research Development (IRAD) programs have confirmedthat more accurate buckling and crippling predictions may be obtained when thecurves (for unidirectional material) are defined in terms of non-dimensionalparameters (Ref 4).” These non-dimensional parameters are displayed in Equation 26.In the following equations and figures Fcc is the crippling stress, Fcu is the ultimatecompression stress of the material, E x is the in plane elastic modulus, E y is thetransverse elastic modulus, b is the width of the loaded edge of the panel, and t isthe thickness of the stiffener. 43
  • ( 26) F cc Ex b E Fcu cu _& _ F E t Ex Ex E y There are two curves provided in Reference 4 dealing with one edge free andno edge free crippling test results. These curves were generated with unidirectionaltape data from the following carbon fiber types:-IM7/5250-4 Data-IM8/HTA Data-Crad AS/3501-6 DataThe one edge free crippling curve is displayed in Figure 2.28. The no edge freecrippling curve is given in Figure 2.29. 44
  • Figure 2.28: One Edge Free Crippling (logarithmic scale) (Ref 4)Figure 2.29: No Edge Free Crippling (logarithmic scale) (Ref 4) 45
  • Crippling Curves 0.8 0.7 0.6 0.5 One edge free Fcc Exc 0.4 No edge Free Fcu E 0.3 0.2 0.1 0 0 2 4 6 8 10 12 bE Fcu tExc Exc E yc Figure 2.30: Crippling Curves The trend line points were taken from Figure 2.28 and 2.29, written down byhand, and were plotted on a linear scaled graph presented in Figure 2.30. Equation 27is used to determine the crippling strength for a one edge free or no edge free longplate. These equations were formulated by adding trend lines to the curves in Figure2.30. Each cap of the stringer in the “I”-section of the wing is modeled as two longplates with one edge free and one edge fixed. The web of the stringer is modeled asone long plate with both edges fixed. The spar caps are modeled as two long plateswith one edge free and one edge fixed. The crippling curves are for one edge free(OEF) and no edge free (NEF) long slender panels. 46
  • The fuselage “Hat”-section cap has one cap plate, two webs, and two flangesas shown in Figure 2.9. The cap is modeled as a long slender plate with both edgesclamped, the webs modeled with both edges clamped, and the flanges with only oneedge clamped. The radius of the curved geometry between the plates is neglected inthe crippling analysis. ( 27) −0.786 F cc Ex ⎛b E Fcu ⎞ (OEF ) = 0.5832 ⎜ ⎟ F cu E ⎜ t Ex Ex E y ⎟ ⎝ ⎠ −0.842 F cc Ex ⎛b E Fcu ⎞ ( NEF ) = 0.9356 ⎜ ⎟ F cu E ⎜ t Ex Ex E y ⎟ ⎝ ⎠ where 12(1 −ν xy 2 ) D11 E= t32.4.3.6.2 Aluminum Equation 28 is used to calculate the crippling strength of the Aluminum AlloyColumns. σ cy is the compressive yield strength. There is a specified cutoff stress for7075-T3 Aluminum Alloy given in Equation 29. The cutoff stress is used as thecrippling strength of the column in the case where the calculated crippling strengthfrom Equation 28 exceeds the cutoff stress in Equation 29. 47
  • ( 28) −0.8 σ cc ⎛ σ ⎞ (OEF ) = 0.546⎜ b cy ⎜ t Ec ⎟ ⎟ σ cy ⎝ ⎠ − 0.8 σ cc ⎛ σ ⎞ (NEF ) = 1.34⎜ b cy ⎜ t Ec ⎟ ⎟ σ cy ⎝ ⎠ (Ref 2) ( 29) σ co = 1.075σ cy (Ref 2)2.4.3.7 Effective Width The concept of effective width is used in determining the buckling stability ofa stiffened panel. The formula used to determine effective width is given in Equation30. The effective width is used to determine the amount of skin area that shares thesame stress as the stiffener before the stiffener buckles. Therefore it is used inconjunction with Euler column buckling to determine the global buckling strength ofthe stiffeners. Figure 2.31 illustrates the concept of effective width. K = 0.85 is usedwhen both edges of the skin is clamped. A plot used to determine the radius ofgyration for a given effective width contribution is given in Appendix B, Figure B.5. 48
  • ( 30) E skin 1we = t skin K E stiffener σ skin (Ref 2) 2we Figure 2.31: Skin Buckled Stiffened Panel and Effective Width Illustration2.4.3.8 Euler Column Buckling Equation 31 is used to determine Euler Column Buckling strength. Le is theeffective length, ρ is the radius of gyration, and the ratio of the two is theslenderness ratio. If the slenderness ratio is below a critical value displayed in 49
  • Equation 31 than inelastic buckling occurs; if the slenderness ratio is above thecritical value than the buckling is elastic. The inelastic portion is referred to as theJohnson Curve, and the Elastic the as Euler curve. The Elastic Modulus in Equation31 corresponds to that of the stringer material. There is a Johnson and Euler ColumnCurve illustrated in Figure B.4 in Appendix B. ( 31)Le ⎞ 2 ⎟ =π ⎟ρ ⎠ crit σ cc E ⎡ 1 σ cc ⎛ Le ⎞ 2 ⎤ L L ⎞σ cr = σ cc ⎢1 − ⎜ ⎜ ⎟ ⎟ ⎥ for e < e ⎟ ⎟ ⎢ 4π E ⎝ ρ ⎣ 2 ⎠ ⎥ ⎦ ρ ρ ⎠ crit π 2E L L ⎞σ cr = for e > e ⎟ (Le ρ ) 2 ρ ρ ⎟ crit ⎠ 50
  • 2.5 Materials Two materials are selected to be representative for this comparison. Thecomposite lamina properties are taken from Reference 4. The lamina and laminatetension ultimate, and open hole strengths, are adjusted to mimic the tensioncompression strength ratio’s that are currently being used in the commercial industry.2.5.1 CFRP Table 2.12: CFRP Lamina Properties o E1 25 Msi α1 0.3 με/C με/C o E2 1.7 Msi α2 19.5 G12 0.65 Msi ν 0.31 ρ 0.056 lb/in3 F1tu 165 ksi F1cu 110 ksi F2tu 4 ksi F2cu 20 ksi F12su 9 ksi Vf 0.6 tp 0.0052 in 51
  • Table 2.13: CFRP Laminate Properties [25/50/25] Ex 9.4 Msi Ey 9.4 Msi νxy 0.32 σtu 60 ksi σoht 48 ksi σcu 40 ksi σohc 27 ksi τsu 24 ksi [50/40/10] Ex 14.5 Msi Ey 5.9 Msi νxy 0.30 σtu 96 ksi σoht 77 ksi σcu 64 ksi σohc 43 ksi τsu 24 ksi [10/80/10] Ex 5.8 Msi Ey 5.8 Msi νxy 0.59 σtu 38 ksi σoht 30 ksi σcu 25 ksi σohc 17 ksi τsu 36 ksi2.5.2 CFRP Lamination Scheme of Aircraft Structure The laminate families chosen for the sub-structure are those that are currentlybeing used in the medium jet transport commercial industry. The actual laminate forthe stringers is not [50/40/10] but rather quasi-isotropic and capped with uni-directional zero lamina plies. When a weighted average of the stiffness of the stringer 52
  • material is taken it is similar, though a little larger, than the skin. For simplicity thestringer family is assumed to be that of the skin.The laminate families for the wing structure are:WingSkin-50/40/10Stringers-50/40/10Spar Caps-10/80/10Spar Webs-10/80/10Ribs (not of significance in this analysis)*The rib spacing is analogous to the stiffened panel length presented in section 2.2.1(wing). The rib pacing is a multiple of the stringers spacing and is maintained aconstant to what is being used in industry.The laminate families for the fuselage structure are:FuselageSkin-25/50/25Stringers-25/50/25Frames (not of significance in this analysis)*The frame spacing is analogous to the stiffened panel length presented in section2.2.2 (fuselage). The frame spacing is a multiple of the stiffener spacing and ismaintained a constant to what is being used in industry. 53
  • The minimum gauge for the quasi-isotropic laminate is 0.055 in. This is theminimum gauge value used in industry, therefore the ply thickness is scaled from thisvalue to determine buildable laminate thicknesses. The minimum gauge laminatethickness corresponds to a ply value of 0.0068 in., different from the ply value used todetermine laminate mechanical properties displayed in Table 2.12.2.5.3 Aluminum The Aluminum Alloy used in this comparison is 7075-T6 sheet properties.The same mechanical properties are used for the sheets and longitudinal extrusions,which actually have slightly different property values. 7075-T6 Aluminum Alloy isnormally used in compression critical areas like the belly of the fuselage and the topof the wing. The Aluminum Alloy does not have good toughness and damage tolerantproperties. The material properties of 7075-T6 Aluminum Alloy sheet are given inTable 2.14. 54
  • Table 2.14: 7075-T6 Aluminum Alloy Properties Alloy Current Yield Strength (ksi) 71 Ultimate Strength (ksi) 80 Fatigue Strength (ksi) 23.7 Elastic Modulus (msi) 10.5 Shear Modulus (msi) 3.9 Shear Stregth (ksi) 48 The fatigue strength for the 7075 Aluminum Alloy is defined as the detailfatigue rating for a Class 1 Notch with a stress ratio of R = 0.06 at 100,000 cycles.The fatigue strength of 24 ksi is used as an estimate based on one third of the yieldstress. The fatigue strength is the same for all tension critical sections of the aircraftexcept for fuselage section tension critical from load case 2 (negative 2.0g HardLanding). The yield strength in Table 2.14 refers to compression yield strength andthe ultimate strength corresponds to ultimate tension strength. The density of 7075-T6is 0.101 lb/in3.2.5.4 Material Enhancement Analysis Cases Each material case is an enhancement of the baseline material. The percentenhancement is chosen based on the feasibility of the enhancement in industry.Technically this is not a sensitivity study because the percent enhancements are notadjusted to give the same unit change in thickness Δt of the structure. A sensitivitystudy would give better weight or volume estimates of the structure that is governed 55
  • by a certain failure mode. Listed below are the material enhancements for thiscomparison.CFRPImproved Composite+25% OHC+25% OHT+50% Elastic ModulusAluminumAdvanced Alloy+10% Elastic Modulus+50% Fatigue Limit+100% Fatigue Limit 56
  • 2.6 Failure Modes and Criteria2.6.1 CFRP2.6.1.1 Wing There is a stiffness redistribution factor applied to the directionalized wingskin. Since the wing box sub-structures are different families of laminates they havedifferent Modulus of Elasticity. For example the [50/40/10] wing skin will take moreloading from wing flexure than the [10/80/10] spar caps. The wing skin has morefiber lamina with zero orientation in the plane of the wing flexure load and thereforehas a larger stiffness in that direction. The stress transformation factor was found inReference 5. The equations and illustrations of the load redistribution are given inAppendix C, Figure C.3.2.6.1.1.1 Fracture The Ultimate Tension and Compression Fracture Strength used for the currentcomposite material and for three improved material cases for Quasi-Isotropic materialis given in Table 2.13. Fracture takes place at ultimate load for all sub-structure.Ultimate load is the limit or design load multiplied by a safety factor of 1.5. Equation32 shows the margin of safety used to size the structure. The margin of safety must begreater than or equal to zero. In the enhancement analysis cases it should be notedthat although the OHC fracture strength is larger for the case of improved composite 57
  • 1, the compression ultimate allowable is not changed, this is also true for OHT. Thefailure criteria for the CFRP laminate wing structure:1) Open Hole Compression (OHC) of skin and spar caps at ultimate load.2) Open Hole Tension (OHT) of skin and spar caps at ultimate load.3) Tension ultimate of stringers at ultimate load.4) Compression ultimate of stringers at ultimate load.5) Shear ultimate of spar web at ultimate load. ( 32) σ Fracture _ AllowableM .S . Fracture = −1 ≥ 0 1.5σ Limit2.6.1.1.2 Buckling1) Compression buckling of stringers at ultimate load.2) Compression crippling of spar caps at ultimate load.3) Combined compression and shear buckling of skin at ultimate load.4) Combined shear and bending buckling of spar webs at ultimate load. 58
  • 2.6.1.2 Fuselage2.6.1.2.1 Fracture1) Open Hole Compression OHC fracture at limit load (when the minimum principal stress in the skin σ2 exceeds the open hole compression strength of the material).2) Open Hole Tension fracture OHT at limit load (when the maximum principal stress in the skin σ1 exceeds the open hole tension strength).3) Ultimate shear fracture at ultimate load (where the maximum shear stress τmax exceeds the shear strength of the skin material)4) Ultimate tension fracture at ultimate load (when axial tension stress σx(+) exceeds the tensile strength of the stringer material)5) Ultimate compression fracture at ultimate load (when the axial compressive stress σx(-) exceeds the compressive strength of the material)2.6.1.2.2 Buckling1) Buckling from combined shear and compression at limit load ( when the axial compression stress σx(-) and the shear stress τxy interacts and makes the skin unstable)2) Buckling from combined shear and tension at limit load (when axial tension stress σx(+) and shear stress τxy interact and the skin becomes unstable) 59
  • 3) Compression Buckling/Crippling at ultimate load (when the axial compression stress σx(-) exceeds the value at which the stringer integrated stiffened panel becomes unstable)2.6.2 Aluminum Alloy Additional Aluminum Alloy properties are given in Appendix C, Figure C.1.Aluminum Failure modes are the same for both wing and fuselage.2.6.2.1 Fracture1) Compression yield of skin, stringers, spar caps at ultimate load.2) Tensile ultimate of skin, stringers, spar caps at ultimate load.3) Shear ultimate of skin and spar web at ultimate load.2.6.2.2 Buckling1) Compression buckling of stringers at ultimate load.2) Compression crippling of spar cap at ultimate load.3) Combined compression and shear buckling of skin at ultimate load.4) Combined shear and bending buckling of spar webs at ultimate load.2.6.2.3 Fatigue1) Fatigue from tension of skin, stringers, and spar caps at limit load, for both loadcases. 60
  • 2.7 Optimization2.7.1 Optimized Variables2.7.1.1 WingOptimized variables include the thickness of all substructures such as:1) Skin thickness (top and bottom)2) Spar web thickness (forward and aft)3) Spar cap thickness (forward and aft)4) Stringer cap/web thickness (top and bottom)2.7.1.2 Fuselage1) Skin thickness (top and bottom)2) Stringer cap/web/flange thickness2.7.2 Fixed Variables1) All material properties are fixed.2) The configuration geometry is fixed (i.e. stiffener spacing and widths of stiffener cross section)* The structural configuration geometry is fixed because otherwise the optimization would be under-constrained.2.7.2.1 WingFixed variables for geometry of the wing box substructure: 61
  • 1) Skin width or wing box width (top and bottom)2) Spar web width or wing box height (forward and aft)3) Spar cap width (forward and aft)4) Stringer width (top and bottom)5) Stringer height (top and bottom)6) Stringer Spacing7) Frame Spacing8) Stringer Geometry2.7.2.2 FuselageFixed variables fuselage structure:1) Fuselage Diameter2) Fuselage Length3) Stinger Spacing4) Frame Spacing5) Stringer Geometry 62
  • 2.7.3 Concept of Minimum Gauge Minimum gauge is the minimum skin thickness for the fuselage that is drivenby a default requirement. The minimum skin thickness is maintained when there is nocritical structural requirement that drives the skin thickness. Default requirements thatdetermine minimum gauge can consist of:1) Structural integrity after impact2) Lightning strike3) Pressurization (or pillowing)4) ManufacturabilityIn this study the Minimum Gauge is 0.055 in. which is the thinnest Quasi-Isotropic([25/50/25]) laminate that can be manufactured.2.7.4 Discretization Figure 2.35 illustrates the sequence of the methodology in the calculationprocess. Figure 2.35 also shows and defines the input parameters for each function inthe methodology. This is in iterative calculation process, therefore the flow chartrepresents an iteration starting from the initial thickness and finishing with theoptimized thickness. The optimized thickness is then put in as the initial thickness forthe next iteration. 63
  • 2.7.4.1 Geometry and Cross Sectional Area Distribution The structure is broken up into sections with “lumped” area a i . The areas arereferred to as “lumped” because they include the cross sectional area of the skin andstringers for the i th section. t is an array of the thicknesses of the structure to beoptimized. bi is a n × 1 array of the widths of each i th panel for a total of n panelsfor the stringers. For example an “I-section stringer” has three panels consisting oftwo caps and a web. The input parameters for the area distribution of the wing aregiven in Equation 33 and for the fuselage, Equation 34. h and w correspond to thewing box height and width respectively. D is the fuselage diameter. Figure 2.32gives an example of how the area of the wing box is implemented as an areadistribution in MATLAB. ( 33) ai = a(h, w, t , bi ) y i = y (h, w, ai ) ( 34) a i = a( D, t , bi ) y i = y ( D) 64
  • ytsk ytsc ytst y z Figure 2.32: Lumping Areas on the Wing Box Once the lateral area distribution is known the center of area y G (or referredto as center of mass) and the second moment of area I z are calculated. The inputparameters for this calculation are given in Equation 35. The discretized formulas inEquation 14, section 2.4.1, are used to calculate the center of area and second momentof area. ( 35) y G = y G (ai , y i ) I z = I (ai , y i , y G ) 65
  • 2.7.4.2 Internal Loads The internal loads are found using the maximum gross takeoff weight,geometry, and load cases from the wide body aircraft. This internal loading is scaledto the weight and geometry of the aircraft being analyzed. The input parameters forthe internal loads on the aircraft structure are given in Equation 36. WGTO is themaximum gross takeoff weight of the aircraft. L is the length of the structure (i.e. thespan of the wing or length of the fuselage). n z is the load factor for the specified loadcase where LC1 is load case 1 and LC2 is load case 2. ( 36) V y = V (WGTO , L, n z _ LC1 , n z _ LC 2 ) M z = M (WGTO , L, n z _ LC1 , n z _ LC 2 ) 66
  • 2.7.4.3 Internal Stress At this point in the method it is required that the user define materialproperties. E skin and E stringer are applied to distinguish between the skin and thestringers, based on their material stiffness. This enables instances where the skin andstringers have different stiffness and therefore carry different axial stress. The input parameters used to calculate internal stress are presented inEquation 37 and 38 for the wing and fuselage respectively. p is the internal cabinpressure. σ xi is the axial stress in the i th lumped area. σ yi is the transverse stress,which is entirely from hoop stress due to pressurization. Axial tensile stress frompressurization is applied to sections in tension from bending, or that are notcompression critical. q i is the shear flow. τ xyi is the shear stress. σ 1, 2i is themaximum and minimum principal stress. ( 37) σ xi = σ x ( M z , I z , y i , y G , E Skin , E Stiff , h, w, t ) qi = q (V y , I z , ai , y i ) τ xyi = τ xy (qi , t ) σ 1, 2i = σ 1 (σ xi , σ yi ,τ xyi ) 67
  • ( 38) σ xi = σ x ( M z , I z , y i , y G , E Skin , E Stiff , P, D, t ) σ yi = σ y ( P, D, t ) qi = q (V y , I z , ai , y i ) τ xyi = τ xy (qi , t ) σ 1, 2i = σ 1 (σ xi , σ yi ,τ xyi )2.7.4.4 Buckling Stability The material properties defined are now used to determine buckling stability.As described the fuselage is viewed as a series of flat stiffened panels around thecircumference of a circle. ν is Poisson’s ratio. b p is the width of the stiffened panel.a p is the length of the stiffened panel. y bi is the local lateral position of each panel ofthe stiffener and is used to calculate the local second moment of area for the stringercross section. The use of these parameters in determining the buckling stability isillustrated in Equation 39. ( 39) σ Cr _ Stiff = σ Stiff ( E Stiff , E Skin, bi , y bi , t ) σ Cr _ Skin = σ Sk ( E Skin , b p , a p , t , σ Cr _ Stiff ,υ ) τ Cr _ Skin = τ Sk (E Skin , b p , a p , t ,υ ) 68
  • 2.7.4.5 Margins of Safety All the information has been calculated to determine the margins of safety foreach i th lumped area. The margins of safety for fracture at limit and ultimate load,buckling at limit load and ultimate load for the wing, and fatigue at limit load arecalculated. These margins of safety are stored in n × 1 arrays. Since the load cases arelaterally symmetric only the margins of safety along the lateral plane are necessary. The fracture failure mode corresponds to Open Hole fracture for a quasi-isotropic composite at limit and the material ultimate fracture strength at ultimate loadfor aluminum. The buckling failure mode corresponds to thin plate buckling for thefuselage skin at limit and Euler beam elastic and inelastic buckling for the stringers atultimate. The fatigue failure mode corresponds to Aluminum Alloy skin at limit loadonly. The margins of safety now get filtered for the most critical one, for eachfailure mode, for the fuselage sections crown and belly. This is done because themargin of safety for the skin and stringers are applied to a variable constraint in theoptimization. Each skin and stringer section has one linear and one non-linearconstraint. 69
  • 2.7.5 Optimization The inputs used in the MATLAB formulated optimization function fminconare illustrated in Equation 40. t is the optimized thickness of the sub-structure. ( 40) t = fmincon ( f ,[ A],[UB ],[ LB ],[C ], t0 ) f min con2.7.5.1 Cost Function f is the cost function which, like the non-linear inequality constraints, isdefined as a separate MATLAB function of the optimized variable t . This is becauseany function where the first or second derivative is used by the optimization isrequired by MATLAB to be defined as a separate “called” function. The cost functionis the function minimized during the optimization process. In this optimization thecost function is the total cross sectional area as a function of the skin and stringerthicknesses. The cross sectional area is used as the cost function because it is directlyproportional to weight. In Equation 41 and 42 A is the area of the sub-structure. ( 41) Awing = Askin + AStringers + ASpar _ Caps + ASpar _ Webs ( 42) A fuselage = Askin + AStringers 70
  • 2.7.5.2 Linear Inequality Constraints Each fuselage sections margins of safety is filtered down to the most critical.This is applied to one linear and non-linear constraint for each variable in thethickness array t . The optimization starting position is t 0 and is the same as theinitial thickness used to determine the initial margins of safety. The starting positionof the optimization never changes only the constraints change. [A] is a 4 × 4 matrixdefining the linear inequality constraints of the optimization. The linear inequalityconstraints take the following form given in Equation 43. {b} is the value that theinequality should not exceed. ( 43) [A] ≤ {b} It is more convenient to apply the margins of safeties directly to theconstraints rather than deriving a constraint directly from the margin of safetyequation and making it a function of thickness directly. A logical process of theapplication of the margins of safety to the constraints is described below. An exampleis presented using the Open Hole Compression (OHC) margin of safety for the quasi-isotropic skin material. t 0i is the initial thickness of the substructure. t fi is theoptimized thickness of the substructure. The OHC margin of safety is given in 71
  • Equation 44. The safety factor S.F. for the fuselage is 1.0 for the fuselage and 1.5 forthe wing for OHC. ( 44) σ OHC M .S . = −1 ( S .F . ) σ 2 If the M .S . < 0 then the minimum principal stress of the skin is larger inmagnitude than the open hole compression strength of the skin material, so the skinfails. Therefore the skin needs to become thicker. Axial stress is inversely related tosecond moment of area (i.e. σ x = − Mc I ). The second moment of area is linearlyrelated to thickness (i.e. I = πD 3 t 8 , for a thin walled cylinder). Therefore axial stressis inversely related to thickness. Assume we have an initial margin of safety ofM .S . 0 < 0 . At what thickness increase would it take to make the margins of safetylarger than or equal to zero. From the relationships established above the requiredthickness can be approximated from Equation 45. The general relationship betweenaxial stress and skin thickness is presented in Figure 2.33. 72
  • σx t Figure 2.33:Generic Axial Stress and Skin Thickness Relationship ( 45) σ OHC t fi M .S . f = −1 ≥ 0 σ 2 t 0iWhere the optimized thickness t fi > t 0i . The linear fracture constraint, with OHC asthe critical failure mode, is given in Equation 46. A similar constraint formulation isvalid for stringer ultimate tensile or compression failure. ( 46) t fi g ( x) Linear = 1 − ( M .S .0 + 1) ≤0 t 0i 73
  • 2.7.5.3 Non-linear Inequality Constraints [C ] is the non-linear inequality constraint which is actually defined, like thecost function, as a separate MATLAB function. The variable being optimized, in thiscase t , is an input parameter. The non-linear inequality constraints have to be equalto or less than zero. ( 47) [C ] ≤ 0 The buckling margin of safety combines compression and shear buckingeffects. The compression and shear buckling stability varies with the skin or webthickness squared (i.e. σ cr , τ cr = const × t 2 ). The relationship between actual axialstress and thickness is also combined to give the cubic relationship in Equation 48.The nonlinear constraint is constructed using the same logical method as the linearconstraint. A generic plot illustrating how axial stress and buckling stability vary withskin or web thickness is given in Figure 2.34. 74
  • σx Axial Stress σx Buckling Strength σcr t Figure 2.34: Generic Buckling Stability and Axial Stress vs Skin Thickness Relationship ( 48) 3 t fi g ( x) non −linear = 1 − ( M .S .0 + 1) 3 ≤0 t 0i Stringer buckling is a methodical procedure. Therefore, the buckling stabilityof a stringer or spar cap does not have an as apparent relationship to flange and webthickness, as it has for the skin thickness. Through a trial and error process the non-linear buckling constraint of the stringers yields the following relationship inEquation 49. This relationship was found by arbitrarily changing the exponent of thethickness ratio until the stringers thickness converged to a margin of safety of zero. 75
  • ( 49) t fi g ( x) non −linear = 1 − (M .S .0 + 1) ≤0 t 0iThere are no equality constraints used in this optimization.2.7.5.4 Upper and Lower Boundaries[UB] and [LB ] are the upper and lower bound limits respectively. The optimizedvariable cannot exceed the upper bound limit values and cannot fall below the lowerbound limit values. The lower bound limit is useful for enforcing minimum gaugerequirement. 76
  • User Defined Fuselage Geometry User Defined Material Properties User Defined Stiffened-Fuselage Diameter D -Stiffness of Skin Eskin Configuration Geometry-Fuselage Length L -Open Hole Compression Fracture -Stiffener Section Length bi Strength (for composite only) σohc -Stiffener Sections Lateral -Open Hole Compression Fracture Position ybiUser Defined Total Airplane Strength (for composite only) σoht -Width of Stiffened Panel bpMaximum Gross Takeoff Weight -Fracture Ultimate Compression -Length of Stiffened Panel ap-Gross Takeoff Weight WGTO Strength σcu -Fracture Ultimate Tensile Strength σtuUser Defined Load Factors User Defined Initial Thickness-Load Factor for Load Case 1 nz_LC1 -Compression yield Strength of Array t0 for-Load Factor for Load Case 2 nz_LC2 Skin (for metallic only) σcy -Crown Skin -Tensile yield Strength of Skin -Crown Stiffener (for metallic only) σty -Belly SkinUser Defined Internal Cabin Pressure -Stiffness of Stringer Estiff -Belly Stiffener-Cabin Pressure P -Poissons Ratio ν Area Distribution and Lateral Position a i = a ( D , t, bi ) yi = y(D ) Centroid and Second Moment of Area Fuselage Internal Load y G = y G (a i , y i ) Skin and Stiffener Compression and V y = V (WGTO , L, n z _ LC 1 , n z _ LC 2 ) I z = I (ai , y i , y G ) Shear Buckling Stability Critical Stress Allowables M z = M (WGTO , L, n z _ LC 1 , n z _ LC 2 ) σ Cr _ Skin = σ Sk (ESkin, bp , a p , t,σ Cr _ Stiff ,υ) σ Cr _ Stiff = σ Stiff (EStiff , ESkin,bi , ybi , t) Axial,Transverse, Shear, and Principle τ Cr _ Skin = τ Sk (ESkin, bp , a p , t,υ) Stresses σ xi = σ x ( M z , I z , y i , y G , E Skin , E Stiff , P , D , t ) σ yi = σ y ( P , D , t ) q i = q (V y , I z , a i , y i ) τ xyi = τ xy (q i , t ) t0 = t ′ σ 1, 2 i = σ 1 (σ xi , σ yi ,τ xyi ) Margins of Safety for all area sections ai -Margin of Safety Array for Fracture of Skin at Ultimate and Limit Load M.S.Fract_sk -Margin of Safety Array for Fracture of Stringer at Ultimate Load M.S.Fract_st -Margin of Safety for Skin Buckling Array M.S.Buck_sk -Margin of Safety for Stringer Buckling Array M.S.Buck_st -Margin of Safety Array for Fatigue Failure of Skin M.S.Fatigue User Defined Optimization Filter for Most Critical Margins of Safety for the Crown (Top) and Belly Starting Thickness t0 (Bottom) -Crown Skin Fracture or Fatigue M.S.Fract_sk_Top -Belly Skin Fracture or Fatigue M.S.Fract_sk_Bottom User Defined Optimizatoin Boundaries -Crown Stringer Fracture M.S.Fract_st_Top -Upper Boundary [UB] -Belly Stringer Fracture M.S.Fract_st_Bottom -Lower Boundary [LB] -Crown Skin Buckling M.S.Buck_sk_Top -Belly Skin Buckling M.S.Buck_sk_Bottom *Boundaries are used when imposing a -Crown Stringer Buckling M.S.Buck_st_Top minimum gauge. -Belly Stringer Buckling M.S.Buck_st_Bottom *The fatigue margin of safety if grouped with the fracture margin of safey for the skin to simplify filtering of critical margins of safety used for linear constraints. the optimization process. in Constraints for Optimizatoin -Linear Inequality Constraints [A] -Non Linear Inequality Constraints [C] *There are no equality constraints. Optimization Cost (Total Cross Sectoinal Area) Optimizing Thickness f = f (t , D , bi ) t = f min con ( f ,[ A],[UB ],[ LB ],[C ], t 0 ) Figure 2.35: Optimization Flow Chart (example for fuselage) 77
  • 3 Verification3.1 Regression Analysis Reference 1 takes a regression approach to verify the load bearing weightcalculation of their model. Reference 1 breaks down the structural weight into threecategories: load bearing, primary, and total structural weight. The substructureincluded in these three categories is listed below (Ref 1). These categories includesubstructure for both the fuselage and the wing.Load Bearing Structural Weight (Ref 1)-Skin-Stringers-Frames-BulkheadsPrimary Structural Weight (Ref 1)-Joints-Fasteners-Keel Beam-Fail Safe Strap-Flooring and Flooring Structural Supplies-Pressure Web 78
  • -Lavatory Structure-Galley Support-Partitions-Shear Ties-Tie Rods-Structural Firewall-Torque Boxes-Attachment FittingsThe total structural weight accounts for all structural members in addition to primarystructural weight. Total structural weight does not include (Ref 1):-Seats-Lavatories-Kitchen-Stowage and Lighting-Electrical Systems-Flight and Navigation Systems-Cargo Commodities-Flight Deck Accommodations-Air Conditioning Equipment-Auxiliary Power Systems-Emergency Systems 79
  • Reference 1 uses the weight breakdown of 8 commercial transport aircraft tocompare the calculated weight from its parametric model (PDCYL). Reference 1 usesthe value of the statistical correlation coefficient to verify the structural weight outputof its model. Regression lines are plotted for load bearing, primary, and totalstructural weight. The model presented in this document calculated weight using thesame aircraft data.3.1.1 Wing Table 3.1 shows the actual weight break downs of the wing for the eightaircraft that reference 1 uses to verify its weight calculations it also shows thecalculated weight produced by the model presented in this document. Figures 3.1through 3.4 show the linear regression lines for the data in Table 3.1. The correlationcoefficients are displayed on each plot. Figure 3.1 presents a comparison of calculatedresults and the weight that NASA’s PDCYL model produced. Table 3.1: Wing Regression Analysis Data Aircraft W calc(lbs) W act (lbs) W GTO (lbs) W primary (lbs) W total (lbs) W PDCYL(lbs) B-720 12,177 11,747 222,000 18,914 23,528 13,962 B-727 8,167 8,791 169,000 12,388 17,860 8,688 B-737 4,243 5,414 149,710 7,671 10,687 5,717 B-747 53,545 50,395 833,000 68,761 88,202 52,950 DC-8 18,533 19,130 310,000 27,924 35,330 22,080 MD-11 38,838 35,157 602,500 47,614 62,985 33,617 MD-83 7,796 8,720 140,000 11,553 15,839 6,953 L-1011 30,450 28,355 430,000 36,101 46,233 25,034 80
  • Figure 3.1: Linear Regression Correlation between PDCYL and Calculated ResultsFigure 3.2: :Linear Regression Correlation between Actual Load Bearing Weight and Calculated Results 81
  • Figure 3.3: Linear Regression Correlation between Actual Primary Weight and Calculated Results Figure 3.4: : Linear Regression Correlation between Actual Total Structural Weight and Calculated Results 82
  • 3.1.2 Fuselage Table 3.2 presents the weight breakdown for the fuselage of the eight aircraftdocumented in reference 1. Figure 3.5 compares the calculated data with that produceby NASA’s PDCYL model. The regression lines comparing the calculated weightwith the actual weight are presented in Figures 3.6 through 3.8. The correlationcoefficients are displayed on the plots. Table 3.2: Fuselage Regression Analysis Data Aircraft W calc(lbs) W act (lbs) W GTO (lbs) W primary (lbs) W total (lbs) W PDCYL(lbs) B-720 6,544 9,013 222,000 13,336 19,383 6,545 B-727 5,530 8,790 169,000 12,424 17,586 5,888 B-737 3,693 5,089 149,710 7,435 11,831 3,428 B-747 30,615 39,936 833,000 55,207 72,659 28,039 DC-8 10,056 13,312 310,000 18,584 24,886 9,527 MD-11 21,726 25,970 602,500 34,999 54,936 20,915 MD-83 5,172 9,410 140,000 11,880 16,432 7,443 L-1011 16,401 28,355 430,000 41,804 52,329 21,608 Figure 3.5: Linear Regression Correlation between PDCYL and Calculated Model Results 83
  • Figure 3.6: Linear Regression Correlation between Actual Load Bearing Weight and Calculated Model Results Figure 3.7: Linear Regression Correlation between Actual Primary Weight and Calculated Model Results 84
  • Figure 3.8: Linear Regression Correlation between Actual Total Structural Weight and Calculated Model Results The results calculated by the model presented in this document correlate betterfor the wing than the fuselage. This could be because there are no tail loads applied tothe sized fuselage structure and because curved geometry is not considered in theweight estimate. Including the frame and rib structure in the weight calculationswould give better correlation for the both the wing and fuselage. 85
  • 3.2 Finite Element The following section verifies the in-plane axial stress and shear stress in themodel with finite element.3.2.1 Wing The wing finite element geometry is given in Figure 3.9. The axial load on theFEM simulating the wing box at the root is given in Figure 3.10. The wing skinmodeled as shell elements show stress concentrations at the root of the wing, wherethe skin meets the spar caps. Figure 3.11 shows the shell element ID numbers so theactual axial stress can be looked up in the output file (“f06” file). From Figure 3.10 itappears that the stress from the parametric model lies in between the orange and theyellow region illustrating the large tensile load in the bottom wing skin, blue and lightblue illustrating the large compression load in the top wing skin. The stress due tobending in the top and bottom skin is approximately 43ksi. Figure 3.9: Wing Finite Element Geometry 86
  • Figure 3.10: Axial Stress in SkinFigure 3.11: Top Skin Element ID Numbers 87
  • Figure 3.12: Axial and Shear Stresses in Elements 4213 and 4214 (“F06 file”) Figure 3.13: Axial and Shear Stress from Model It is shown that the stresses in the “f06 file”, Figure 3.12, are approximatelyequal to the stresses given in the model presented in this document, Figure 3.13.These stresses are for elements on the top skin of the wing box root, the element IDnumber are given in Figure 3.11. 88
  • 3.2.2 Fuselage The fuselage model is a simple un-tapered cylinder. The axial stressdistribution is given in Figure 3.14. The axial stress in the model presented in thisdocument is given in Figure 3.15. The “f06 file” is not displayed because the stress inFigure 3.15 is consistent with the stress in the finite element model, Figure 3.14. Figure 3.14: Axial Stress 89
  • Figure 3.15: Axial Stress from Model Figure 3.16 illustrates the shear stress distribution in the finite element model.Figure 3.17 shows the shear stress distribution calculated from the model presented inthis document. The shear stress distribution in the finite element model is consistentwith the shear stress distribution calculated in this model. 90
  • Figure 3.16: Shear StressFigure 3.17: Shear Stress Calculated in Model 91
  • 4 Results and Discussion The baseline and enhanced material case results are presented for CFRP andAluminum Alloy in Table 4.1 and 4.2 respectively. The weight estimate includes themass of the longitudinal load bearing structure, which for the wing and fuselagestructure is the skin and stringers. The percent weight benefit of the enhancedmaterial cases are also presented in Tables 4.1 and 4.2 (next to the calculated weightvalues). The CFRP resulting structural load bearing weight calculations neglectwhether the optimized thickness is realistic to manufacture along with being asymmetric and balanced laminate. The percent weight benefit is calculated inreference to the baseline material. 92
  • Table 4.1: CFRP Baseline and Enhanced Material Weights (% Structural Weight Reduction) Improved Composite 1 Improved Composite 2 Improved Composite 3 Baseline Material Baseline+25%OHC Baseline+25%OHT Baseline+50% Elastic Modulus Wing Structural Weight (lb) Wide Body 16,737 14,829 (11.4%) 15,811 (5.5%) 16,635 (0.6%) Narrow Body 3,177 2,860 (10.0%) 3,011 (5.2%) 3,167 (0.3%) Fuselage Structural Weight Wide Body 10,968 10,554 (3.8%) 10,959 (0.1%) 10,199 (7.0%) Narrow Body w/ 2,337 2,337 (0.0%) 2,337 (0.0%) 2,189 (6.3%)Min Gauge =55mil Narrow Body Min 1,923 1,923 (0.0%) 1,923 (0.0%) 1,713 (10.9%) Gauge=30mil Total Structural Weight (lb) Wide Body 27,705 25,383 (8.4%) 26,770 (3.4%) 26,834 (3.1%) Narrow Body w/ 5,514 5,197 (5.7%) 5,348 (3.0%) 5,356 (2.9%)Min Gauge =55mil Narrow Body 5,100 4,783 (6.2%) 4,934 (3.3%) 4,880 (4.3%)Min Gauge = 30mil The fatigue or strength depends on the material notch toughness, nominalload, structure geometry, notch geometry, detectable crack size, and loading cycleprofile for the aircraft. The fatigue performance for the Aluminum Alloys currentlyused in industry is unknown. Rather enhanced fatigue performance cases are studiedto determine a range of weight benefit from the application of Aluminum Alloy withhigh fatigue performance. 93
  • Table 4.2: Aluminum Baseline and Enhanced Material Weights (% Structural Weight Reduction) Advanced Alloy 1 Advanced Alloy 2 Advanced Alloy 3 +10% Elastic +50%Fatigue +100% Fatigue Baseline Material Modulus Performance (36 ksi) Performance (48 ksi) Wing Structural Weight (lb) Wide Body 33,046 32,941 (0.3%) 25,998 (21.3%) 22,595 (31.6%) Narrow Body 6,187 6,160 (0.4%) 4,915 (20.6%) 4,305 (30.4%) Fuselage Structural Weight Wide Body 16,575 16,560 (0.1%) 12,585 (24.1%) 11,319 (31.7%) Narrow Body 3,749 3,749 (0.0%) 2,688 (28.3%) 2,277 (39.3%) Total Fuselage Structural Weight Wide Body 49,621 49,501 (0.2%) 38583 (22.2%) 33,914 (31.7%) Narrow Body 9,936 9,909 (0.3%) 7603 (23.5%) 6582 (33.8%) To judge the consistency of the method used to determine the stiffener spacingand geometry, the baseline material structural weights are compared to their structuralun-stiffened configuration weight. It is determined that all the structure is stiffnesscritical if un-stiffened. The ratio of the stiffened to un-stiffened weight for the wingand fuselage of the wide and narrow body aircraft is given in Table 4.3. It is desiredthat the numbers between the CFRP and Aluminum Alloy for the wing and fuselageof each aircraft are similar. This helps insure that a weight performance benefit forthe application CFRP over Aluminum is not biased because the CFRP is moreefficient than the Aluminum Alloy stiffened configuration. 94
  • Table 4.3: Stiffened and Un-stiffened Structural Weights Baseline Material Stiffened Baseline Material Un-Stiffened Stiffened/Un-Stiffened Configuration Weight (lb) Configuration Weight (lb) Weight (lb/lb) Wide Body Wing CFRP 16,601 52,011 0.32 Aluminum 33,046 100,509 0.33Fuselage CFRP 10,538 26,610 0.40 Aluminum 16,575 44,210 0.37 Narrow Body Wing CFRP 3,147 12,515 0.25 Aluminum 6,187 24,178 0.26Fuselage CFRP (MG = 55mil) 2,229 7,580 0.29 Aluminum 3,749 12,868 0.29How does the weight performance benefits from the application of CFRP andAluminum Alloy on a medium transport aircraft compare with that on a smalltransport aircraft? The results indicate that the small transport aircraft will have an almost equalstructural weight benefit that the medium aircraft has using CFRP compared withAluminum Alloy. Figure 4.1 plots the weight benefit of CFRP compared toAluminum Alloy for the narrow and wide body aircraft for a range of fatigueperformance. If the Aluminum wide and narrow body aircraft have different fatigueperformance behavior they will see different benefits from the application of CFRP astheir primary load bearing structural material. When the minimum laminate thicknessis found for each structural region, allowing a 5% variation in the target laminatefamily (i.e. for the fuselage [25/50/25]), a reduction is seen in the weight benefit of 95
  • CFRP when compared to Aluminum Alloy. The decrease is more dramatic in the caseof the narrow body. Using the ply thickness of the current material used on themedium jet transports, the narrow body will see a weight penalty when compared toan Aluminum Alloy with fatigue strength above 33ksi. Therefore, based on thisanalysis, with the current laminate thicknesses the narrow body will not have theweight benefit from the application of CFRP that the wide body does. Whenneglecting minimum laminate thickness, the narrow body will have a similar weightperformance benefit as the wide body. The total structural weight values of CFRPaircraft is in Table 4.1. and for Aluminum Alloy in Table 4.2. 60.0% WB Percent Weight Benefit 50.0% WCFRP % Re duction = 1 − × 100 NB Percent Weight Benefit MG=55mil 45.9% W Al Percent Weight Benefit with 45.3% Manufacturable Laminate Weight Reduction of CFRP Compared to Aluminum (%) 40.0% Thickness 38.4% 29.7% 30.0% 29.3% 20.0% 20.0% 18.8% 20.8% 18.3% 10.0% 9.9% 0.0% 20 25 30 35 40 45 50 -6.2% -10.0% -20.0% -22.7% -30.0% Fatigue=24ksi Fatigue=36ksi Fatigue=48ksi -40.0% Fatigue Strength (ksi) Figure 4.1: Percent Load Bearing Structural Weight Benifit of CFRP over Aluminum for a Range of Potential Fatigue Performance 96
  • What are the critical structural design drivers with the application of CFRP onthe medium and small commercial transport and how do they compare? The data from Table 4.1 shows that the wide and narrow body CFRP wingexperiences the largest reduction in weight from OHC enhancement (ImprovedComposite Case 1). Therefore OHC is the most weight sensitive design driver for theCFRP wing box structure on both the wide and narrow body aircraft. OHC is acritical design driver because it is the lowest material strength allowable governingthe design. The wide body CFRP wing has more of a weight benefit because it hasthicker skin and a larger load. Enhancing a lower stress allowable yields a largerthickness benefit than increasing high stress allowable this is illustrated in Figure 4.2.This is also the reason why increasing OHC yields more of a weight performancebenefit than enhancing OHT. Figure 4.3 shows the failure mode distribution on thewing structure, which is the same for the wide and narrow body and all subsequentmaterial enhancements except Improved Composite 1 applied to the narrow body.Relationship of thickness change with the change of low and high stress allowables isgiven in Figure 4.2. Figure 4.3 shows an increase in the stiffness critical region on theupper wing skin of the narrow body aircraft with a 25% OHC enhancement. ( 50) 1 t∝ σ Design σ All M .S . = − 1 = 0 → σ Design = σ All σ Design 97
  • Thickness Δt Δt > Δtl Δσ l Δσ h Δth Δσ h Design Stress Δσ l Figure 4.2: Skin Thickness Design Stress Relationship 98
  • Skin OHC Fracture Skin Compression Buckling Spar Web Shear-Bending Buckling Skin OHT Fracture Top Skin Spar Web Bottom Skin % Length 10% 11% 10% 16% 6% 15% 16% 16%Figure 4.3: Failure Mode Distribution on Wing Structure for Wide and Narrow body (CFRP Baseline and all cases except for narrow body Improved Composite 1) 99
  • Skin OHC Fracture Skin Compression Buckling Spar Web Shear-Bending Buckling Skin OHT Fracture Top Skin Spar Web Bottom Skin % Length 10% 11% 10% 16% 6% 15% 16% 16% Figure 4.4: Failure Mode Distribution for Narrow Body Wing (CFRP Improved Composite 1) The most weight sensitive material enhancement for the wide and narrowbody CFRP fuselage is the 50% increase in the Elastic Modulus. Table 4.1 shows thatin the case of the relaxed minimum gauge the narrow body experiences more of aweight benefit than the wide body from the stiffness enhancement. The wide body hasless of a benefit from stiffness enhancement because a portion of its fuselage is OHCfracture critical. If the narrow body fuselage structure is limited by minimum gaugethan the wide body has more of a weight benefit from stiffness enhancement. Both 100
  • aircrafts fuselage structure has OHT fracture critical sections. They have a smallamount of weight benefit from OHT enhancement because, as stated earlier,enhancements for allowables at large stress levels, relative to the structure, have asmaller decrease in thickness or no decrease in thickness if limited by minimumgauge. Figure 4.5 shows the failure mode distribution for the wide body fuselage.Figure 4.6 shows the failure mode distribution for the narrow body fuselage, whichhas the same failure mode distribution for all subsequent material enhancements. Skin OHT Fracture Skin Compression-Shear Buckling Skin/Stringer Compression Buckling Skin OHC Fracture MG Minimum Gauge WB Baseline MG MG WB Improved Composite 1 (+25%OHC) MG MG WB Improved Composite 2 (+25%OHT) MG MG WB Improved Composite 3 (+50%E) MG MG %Fuselage Length 24.5% 12.5% 15.2% 18.0% 12.4% 17.4% Figure 4.5: Failure Mode Distribution on Wide Body Fuselage (CFRP) 101
  • Skin OHT Fracture Skin Compression-Shear Buckling Skin/Stringer Compression Buckling Skin OHC Fracture MG Minimum Gauge NB Baseline MG MG MG MG MG %Fuselage Length 24.5% 12.5% 15.2% 18.0% 12.4% 17.4% NB Improved Composite 1 MG MG MG MG MG NB Improved Composite 2 MG MG MG MG MG NB Improved Composite 3 MG MG MG MG MG MG MG %Fuselage Length 24.5% 12.5% 15.2% 18.0% 12.4% 17.4% Figure 4.6: Failure Mode Distribution on Narrow Body Fuselage (CFRP) The most weight sensitive material enhancement is OHC for both the wideand narrow body aircraft total load bearing structural weight. The second most weightsensitive material enhancement is OHT for the wide and narrow body aircraft Though OHT is neither the CFRP wing or fuselage structures most weightsensitive material enhancement it provides more of a weight reduction in the wing 102
  • than Improved Composite 3 provides in the fuselage structure. If the narrow bodyminimum gauge is relaxed to 30 mils the Improved Composite 3 stiffnessenhancement is the second most weight sensitive material case for the total loadbearing structural weight. There are portions of the fuselage structure that are OHTcritical where the critical stress is hoop stress. The skin on the OHT critical(pressurization critical) portions of the fuselage are so thin that you see little weightbenefit from Improved Composite 2, in the narrow body case there is no weightbenefit because the skin thickness is limited by minimum gauge. The wide and narrow body wing has a similar critical failure mode pattern.Both aircrafts have OHT critical lower skins of the wing and a large percentage of thetop skin OHC critical. The OHC enhancement provided more of a weight benefit thanOHT, even though it’s critical in a smaller portion of the wing, because it’s a muchlower stress allowable therefore thickness reduction is more sensitive to OHCenhancement. The primary failure mode in the wide and narrow body fuselage iscompression-shear buckling. Only one fuselage section of the wide body is OHCcritical, the narrow body has none. There is a large weight benefit in the wide bodyfuselage when the OHC is enhanced because it has the largest reduction in skinthickness. 103
  • What are the critical failure modes on the Aluminum Alloy wide and narrowbody aircraft structure? The wide and narrow body airframe is most sensitive to fatigue performance.The Aluminum Alloy wing and fuselage have a large buckling critical portion ofthere structure but a 10% stiffness enhancement shows very little weight benefit inthe thickness range. The wide and narrow body wing show similar weight benefitswhen enhancing the Fatigue strength. The failure mode distribution is similar for theAluminum Alloy wide and narrow body aircraft along with all subsequent materialenhancements. The failure distribution for the Aluminum Alloy wing structure on thewide and narrow body aircraft is given in Figure 4.7. 104
  • Skin Compressive Yield at Ultimate Skin Compression Buckling Spar Web Shear-Bending Buckling Skin Fatigue Tension Failure Top Skin Spar Web Bottom Skin% Length 10% 11% 10% 16% 6% 15% 16% 16% 0 0.11 0.21 0.37 0.43 0.58 0.74 0.9 Figure 4.7: Failure Mode Distribution for Wide Body and Narrow Body Wing (Aluminum) 105
  • The Aluminum Alloy narrow body fuselage has the most weight benefitbecause it is entirely fatigue critical due to pressurization. Table 4.2 shows weightbenefits of the Aluminum material enhancements which include 10% ElasticModulus, 50%Fatigue performance, and 100% Fatigue performance enhancement.The material fatigue enhancement cases give a range of possible fatigue performancefor an Aluminum Alloy aircraft similar to the size of the narrow body and wide bodyaircraft. Figure 4.8 shows the failure mode distribution for the Aluminum Alloy widebody fuselage and all subsequent material enhancements. Figure 4.9 shows the failuremode distribution on the Aluminum Alloy narrow body fuselage and all subsequentmaterial enhancements. The Aluminum Alloy wide body aircraft with a fatigue performance of 24 ksiis almost entirely fatigue critical from fuselage pressurization and bending tensionstress, except for a fracture critical section above aft of the wing box. The wide bodyAdvanced Alloy 2 optimized failure mode output and state of stress with therespective critical load case is shown in Figure 4.8. Advanced Alloy 3 whichincreases the fatigue performance to 48ksi has a failure mode scheme where the entireforward fuselage is compression and shear buckling critical and the mid-section(above the wing box) is tension fracture critical from load case 2. The failure modesare similar for the narrow body aircraft except there is a larger area of the skin that isfatigue fracture critical from pressurization for each material enhancement case.Therefore the critical failure modes on the wide and narrow body aircraft include: 106
  • 1) Fatigue fracture of skin from tension stress due to pressurization and bending from load case 1 (2.5g maneuver).2) Ultimate tension strength fracture of skin and stringers from bending induced by load case 2 (-2.0g hard landing). Skin Fatigue Tension Failure Skin/Stringer Ultimate Tension Fracture Skin Compression-Shear Buckling Skin/Stringer Compression Buckling WB Baseline WB Advanced Alloy 1 (+10%E) WB Advanced Alloy 2 (+50%Fatigue) WB Advanced Alloy 3 (+100%Fatigue) %Fuselage Length 24.5% 12.5% 15.2% 18.0% 12.4% 17.4% Figure 4.8: Failure Mode Distribution for Wide Body Fuselage (Aluminum) 107
  • Skin Fatigue Tension Failure Skin/Stringer Ultimate Tension Fracture Skin Compression-Shear Buckling Skin/Stringer Compression Buckling MG Minimum Gauge NB Baseline NB Advanced Alloy 1 (+10%E) NB Advanced Alloy 2 (+50%Fatigue) NB Advanced Alloy 3 (+100%Fatigue) MG MG MG MG MG%Fuselage Length 24.5% 12.5% 15.2% 18.0% 12.4% 17.4%Figure 4.9: : Failure Mode Distribution for Narrow Body Fuselage (Aluminum) 108
  • 5 Conclusions The wide and narrow body aircraft yield similar failure mode distributions forCFRP Laminate. The wing structure has the most weight benefit from theenhancement of the OHC allowable and the fuselage structure has the most weightbenefit from the enhancement of stiffness. In the case of an Aluminum Alloy structure, the wide and narrow body resultsshow similar failure mode distributions for the wing and fuselage. The AluminumAlloy Structure is primarily fatigue failure critical and has the largest weightreduction with the enhancement of fatigue performance. Since the wide and narrow body results show similar failure modedistributions it is not the “type” of failure that limits the feasibility of CFRP laminatebeing applied to the narrow body aircraft structure but rather the range of thickness ofthe laminate. With Improved Composite 3 (25% Elastic Modulus enhancement) morethan half of the narrow body fuselage structure was limited by Minimum gauge.Therefore the weight benefit from the application of CFRP laminate on the narrowbody fuselage structure is limited by minimum gauge of the laminate. 109
  • 6 RecommendationsTo improve the accuracy of the optimization model the following could be done:1) Improving the accuracy of the critical fatigue stress (for aluminums) by conducting a more detailed fatigue methodology.2) Study multiple materials for the different sections of the aircraft. This could include different types of aluminums, CFRP laminates, laminas, or fabric for different wing and fuselage sections.3) Studying different structural concepts such as Honeycomb Sandwich and Iso-gridconfigurations.4) Obtain and apply information on criteria for different failure modes such as strength after impact for CFRP, pillowing of fuselage skin due to pressurization, and delaminating due to joint flexure for CFRP.5) Applying methodology for sizing ribs and frames to include in the load bearing structural weight calculation.6) Applying load cases that subject the structure to torsion. 110
  • 7) Use linear regression to scale the takeoff weight of the aircraft to the calculate load bearing structural weight in order to approximate the change in load on the structure.8) Include tail loads to size the fuselage structure.9) Calculate actual producible laminate thicknesses that correspond to ply thickness and laminate family.To simplify the model and reduce computational load the following could bedone:1) Produce a basic axial, transverse, and shear running load profile for the wing and fuselage for a class of aircraft (w/ similar geometry). So that it could be mapped for different size aircraft. The running loads should be specific to different sub- structure.2) Specify reasonable convergence criteria to reduce the number of iterations necessary to obtain an optimum thickness. 111
  • References(1) Ardema M.D., Chambers M.C., Patron A.P., Hahn A.S., Miura H., Moore M.D.,1996. Analytical Fuselage and Wing Weight Estimation of Transport Aircraft, NASATechnical Memorandum 110392(2) Curtis, Howard D. Fundamentals of Aircraft Structural Analysis 1997. Richard D.Irwin, a Times Mirror Higher Education Group, Inc.(3) Raymer, D.P. Aircraft Design: A Conceptual Approach 3rd Ed, 1999. AmericanInstitute of Aeronautics and Astronautics, Inc.(4) MIL-HDBK-17-3D, Polymer Matrix Composites Volume 3. Material Usage,Design, and Analysis 1997, U.S. Department of Defense.(5) Hibbler, Mechanics of Materials Fifth Edition 2003, Pearson Education, Inc.(6) Lan, E.L. Roskam, J. Airplane Aerodynamics and Performance Third Printing,2003. Design, Analysis and Research Corporation (DARcorperation)(7) Bruhn, E.F. Analysis and Design of Flight Vehicle Structures, 1973. JacobsPublishing, Inc. 112
  • (8) Kollar, L.P. Mechanics of Composite Structures, 2003. Cambridge UniversityPress. 113
  • A Appendix Figure A.1: Medium Body Jet Transport A.1
  • Figure A.2: V-N Diagram Specifications for Military Airplanes (Ref 6) A.2
  • B AppendixExample Calculation to Determine Stringer Dimensions and SpacingUsing the running load from load case 1, wing station 66. kipsN − = 9.5 inσ ohc = 43ksi(ultimate) → σ Limit = 29ksi(compression) N−t skin = = 0.331in σ ohcFigure 7.12 (Ewing Buckling Presentation), simply supported boundary conditionsare used. Contribution of effective width is neglected because of the heavy thicknessof the skin (Bruhn). 2 ⎛t ⎞σ cr = KE ⎜ skin ⎟ ⎝ b ⎠a = 2( ss) → K = 3.6bb = t skin σ cr = (0.331in ) (3.6)(14.5msi ) = 14in KE (29ksi )a = 2 → a = 28in.b a 28Le = = = 23in. c 1.5Note: The calculation of the stringer spacing b above is only used to estimate theeffective length of the stiffener column. A slenderness ratio is chosen from the Euler-Johnson Column Curve (Figure 4) for a crippling stress of σ cc = 30ksi .Le = 24 → ρ = 0.958 ρ B.1
  • The following is a derivation to calculate all the solutions for the dimensions of an“I”-beam section type stiffener for a radius of gyration, ρ . 2 1.8 bf 1.6 1.4 bw 1.2 bf (in) 1 0.8 0.6 0.4 0.2 0 2 2.2 2.4 2.6 2.8 3 3.2 3.4 bw (in)The following dimensions were chosen for the “I”-beam section type stiffener.b f = 1.0inbw = 2.4in.Since the crippling stress is known an estimate for the thickness comes from Figure 3.bf bw ≈ 0.4 → = 49 → t stiffener = 0.049inbw t stiffenerAstiffener = (2b f + bw )t stiffener = [2(1.0in ) + 2.4in](0.049in ) = 0.216in 2The following is a method of estimating the contribution of each stiffener to thesecond moment of area. The width of the top skin at W.S. 66 is: B.2
  • ⎛ in ⎞width = (4.077 ft )⎜12 ⎟ = 49in ⎜ ⎟ ⎝ ft ⎠ .n: number of stringersContribution of area for n stiffeners on the tops skin per unit thickness is: ΔA = ( 0.216in 2 n ) ( = 4.4 × 10 −3 in n )width 49inFrom the past analysis it is known that this wing station is open hole compression(top skin) fracture critical. To estimate the number of stringers it is necessary to findthe stress at which the following equations converge. = (29ksi ) t skin 0.331inσx =σ0 t skin + ΔA ( ) 0.331in + 4.4 × 10 −3 in. n width 2 2 ⎛ ⎞ ⎛ 0.331in. ⎞ ⎜ width n ⎟ = 3.6(14.5msi )⎜ 49in. n ⎟ tσ cr = KE ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ B.3
  • Converging Stress 50 45 40 35 Stress (ksi) 30 Axial Stress 25 Critical Buckling Stress 20 15 10 5 0 0 1 2 3 4 5 Number of StringersThe equations above converge between 3 and 4 stringers at a stress of about 28ksi. Anapproximate stringer spacing is between the two calculated values below. width 49in.b1 = = ≈ 16in. n1 3 width 49in.b2 = = ≈ 12in. n2 4b2 ≤ b ≤ b112in. ≤ b ≤ 16in. B.4
  • Wide Body 25 20Running loads (lb/in) 15 10 5 0 0 10 20 30 40 50 60 70 80 90 Wing Station (ft) Figure B.1:WB Wing Running Loads Narrow Body 10 9 8 7Running Loads (lb/in) 6 5 4 3 2 1 0 0 10 20 30 40 50 60 Wing Span (ft) Figure B.2: NB Wing Running Loads B.5
  • 120 100 bf 80 bw bw/t 60 0.1 0.2 0.3 40 0.4 20 0.5 0.6 0.7 0.8 0.9 1.0 bw bf 070 60 50 40 30 20 10 0 Crippling Stress (ksi) Figure B.3: Crippling of CFRP Laminate "I"-Section Beam With Ex=14.5Msi B.6
  • 90 Johnson & Euler Column Curves 80 π 2E Euler Curve σ cr = (Le ρ )2 70 2 ⎡ 1 σ cc ⎛ Le ⎞ ⎤ Johnson Curve σ cr = σ cc ⎢1 − ⎜ ⎟ ⎥ 4π 2 E ⎜ρ ⎟ ⎢ ⎣ ⎝ ⎠ ⎥ ⎦ 60 E=14.5Msi σ cr 50 Ksi 40 30 20 10 Figure B.4: Euler Column Buckling of CFRP Laminate Ex=14.5Msi 0 0 10 20 30 40 50 60 70 80 90 100B.7 Le ρ
  • 3 we S 2.5 C.G. of Stiffener Alone A0: Area of Stiffener Alone ρ0: Radius of Gyration of Stiffener Alone 2 S ρ0 3.0 1.5 2.8 2.6 Width 2.4 1 2.2 2.0 1.8 1.6 0.5 1.4 1.2 1.0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 we t skinB.8 Figure B.5: Radius of Gyration for Euler Buckling Calculation with Contribution of Effective A0
  • C Appendix Figure C.1: 7075 Aluminum Alloy Mechanical Properties (Ref 4) C.1
  • Figure C.2: "K" Values for Compression and Shear Panel Buckling (Ref 7) C.2
  • Example of Composite beam stress correction on Wing BoxSimplified Wing Box (Reference 8) Initial Stress Distribution E1 t1 (-)compression E2 σ t2 (+) tensionWhere E1>E2 E1n= E2t1′ = nt1Making the whole Wing box equivalent area with respect to material 2 t1<t1 Stress Distribution after equated to material 2 t 1 (-)compression t2 σ (+) TensionUpdate centroid with new area distribution y′ = ∑ yA ∑AUpdate moment of inertial about the centroid I ′ = ∑ A′( yi − yc′ ) 2Update the axial stress on top skin (material 1) − M ( yi − yc′ ) σx = ′ I xxApply transformation factor again to get actual stress in top skin (material 1) σ x = nσ x ′σx in this example is the stress in the top skin Figure C.3: Stiffness Correction C.3
  • D AppendixSample CalculationSample Hand calculations are done to clarify the methodology used to size thefuselage structure. Fuselage section 46 is used to verify the optimization calculations.The sample hand calculations are done by treating the fuselage cross section as a thinwalled tube.Geometry and Area DistributionEquivalent Skin Area t crown _ skin = 0.1095in A stringer Vy t crown _ stringer = 0.0300in yG Mz Askin y t belly _ stringer = 0.1106in x z t belly _ skin = 0.1825inFigure D.1: Illustration of Fuselage Modeled as Idealized Tube Fuselage Section 46Figure D.2: Optimized Thicknesses for Fuselage Section 46 D.1
  • An equivalent skin thickness is calculated to take into account the area of the stringersin addition to the skin area. Also, the equivalent skin thickness is calculated to use theequation for the center of mass, first, and the second moment of area that are derivedfrom a thin walled tube. Figure 3.1 illustrates the fuselage analysis with the internalloads applied to a thin walled tube.The stringer panel lengths bi , are given in the calculations below. There areapproximately 30 stringers on each the crown and belly of the fuselage of the wide ′ t eq Abody aircraft. The equivalent skin thickness is . is the calculated Area for thecrown and belly of the fuselage separately.Crownb flange = 0.5inbweb = 1.25inbcap = 1.5inAcrown _ stringers = [2(0.5in ) + 2(1.25in ) + (1.5in )](0.03in )(30 stringers ) = 4.5in 2 4.5in 2t eq _ crown _ stringers = = 0.01256in ⎛12 in ⎞ (19 ft )⎜ ⎝ ft ⎟ ⎠ π 2 ′t eq _ crown = 0.1095in + 0.01256in = 0.1220in 19 ft ⎛12 in ⎞ ⎜ ⎝ ft ⎟ ⎠ (0.1220in ) = 43.69in 2Acrown =π 2 D.2
  • BellyAbelly _ stringers = [2(0.5in ) + 2(1.25in ) + (1.5in )](0.1106in )(30stringers ) = 16.59in 2 16.92in 2t eq _ belly _ stringers = = 0.0463in ⎛12 in ⎞ (19 ft )⎜ ⎝ ft ⎟ ⎠ π 2 ′t eq _ belly = 0.1825in + 0.0463in = 0.2288in 19 ft ⎜12 in ⎞ ⎛ ⎝ ft ⎟ ⎠Abelly =π (0.2288in ) = 81.94in 2 2Center of Mass r y θFigure D.3: Lateral Cut of Fuselage as a Idealized TubeBelow the first and second moment of area is derived assuming the area equation of athin walled tube. The radius of a lateral cut of the fuselage is r , and is illustrated inFigure 3.3. θ is the angle from the center of the circle as illustrated in Figure 3.3. y D.3
  • is the lateral distance from the center of the circle. The First Moment of Area (i.e.∫ ydA ) is used to find the center of mass and the shear flow at the center of the ∫y 2 dAfuselage. The Moment of Inertia I zz , is the Second Moment of Area (i.e. ).Figure 3.4 gives the center of mass given by the MATLAB code. 61 1 ∑ya i iy g = ∫ ydA = i =1 61 ∑a A i i =1 π π∫ ydA = ∫ (r sin θ )(πrt )dθ = r t (1 − cosθ ) = 2r t 2 2 0 0Ahalf _ tube = πrt 2r 2 t 2 2 (19 ft )(12 in ft )y g _ half _ tube = = r= = 72.57in πrt π π 2yg = ( Acrown y g _ crown + Abelly y g _ belly = ) 43.69in 2 − 81.94in 2 (72.57in ) = −22.09in Acrown + Abelly 43.69in 2 + 81.94in 2Figure D.4: Center of Mass (in)Second Moment of AreaThe equation derived for the second moment of area assumes a thin walled tube ofuniform thickness with its center of mass in the center of the circle. Since the actualcenter of mass is not in the center of the circle the equation for the second moment of D.4
  • ( )area needs to be uncoupled using the parallel axis theorem (i.e. I x + A y − y c ). 2Figure 4.5 shows the second moment of area calculated by the MATLAB code.I zz = ∫ y dA = ∑ ai ( y i − y g ) 61 2 2 i =1 2π πD 3t∫y dA = ∫ (r sin θ ) rtdθ =πr t = 2 2 3 0 8 ′ ′ ⎛ t eq _ crown + t eq _ belly ⎞ ⎜ πD 3 ⎜ ⎟ ⎟ ⎛A ⎠ − ⎜ crown + Abelly ⎞ ⎟(0 − y g ) =I zz = ⎝ 2 ⎟ 2 ⎜ 8 ⎝ 2 ⎠ ⎛ 0.1220in + 0.2288in ⎞π ⎢(19 ft 3 )⎛12 in ft ⎞⎤ ⎜ 3 ⎡ ⎜ ⎟⎥ ⎟ ⎣ ⎝ ⎠⎦ ⎝ ⎠ − ⎛ 43.69in + 81.94in ⎞ 2 2 ⎟(0 − 22.09)2 = 785,731in 4 2 ⎜ ⎜ ⎟ 8 ⎝ 2 ⎠Figure D.5: Second Moment of Area (in4)Internal LoadsThe optimized structure of fuselage section 46 is driven by load case 2 (-2g hardlanding). The shear and moment values in fuselage section 46 are given below andare verified in the MATLAB code in Figure 3.6.V y = 5.92 × 10 5 lbsM z = −2.725 × 10 8 lbs ⋅ in D.5
  • Figure D.6: Internal Loads (shear(lbs), moment(lbs*in))Internal StressMaximum Principle Stress in Fuselage CrownThe skin on the crown of fuselage section 46 is OHT critical, therefore the maximumprinciple stress in the crown of the fuselage is calculated below. It is assumed that thefailure takes place at the top of the crown where the axial stress is the greatest. Thetop of the crown there is no shear stress so the max principle stress equals the axialstress and the minimum principle stress is the transverse tension stress frompressurization. Figure 3.7 shows the principle stresses given in the MATAB code. ⎛ 19 ft × 12 in ⎞ ⎜ ft ⎟ (18 psi )⎜ ⎟ ⎜ 2 ⎟σ x _ pressure = Pr = ⎝ ⎠ = 9,370 psi 2t 2(0.1095in ) ⎛ 19 ft × 12 in ⎞ ⎜ ft ⎟ (18 psi )⎜ ⎟ ⎜ 2 ⎟σ y _ pressure = Pr = ⎝ ⎠ = 18,740 psi t (0.1095in ) D.6
  • ⎛ 19 ft × 12 in ⎞ ⎜ ⎟ ( ) − − 2.725 × 10 lb ⋅ in ⎜ 8 ft − (− 22.09in )⎟ − M zc ⎜ 2 ⎟σ x _ bending = = ⎝ ⎠ = 47,197 psi 4 I 785,731inσ x = σ x _ pressure + σ x _ bending = 9,370 psi + 47,197 psi = 56,567 psiσ y = σ y _ pressure = 18,740 psiσ 1 = σ x = 56.57ksiσ 2 = σ y = 18.74ksi σ1 σ2Figure D.7: Maximum and Minimum Principle Stress in Crown (psi)Minimum Principle Stress in Fuselage BellyThe minimum axial compression stress takes place at the very bottom of the belly ofthe fuselage. At the bottom of the fuselage there is no in-plane shear stress andpressure effects are not taken into account in the axial direction. Since there is noshear stress at the bottom of the fuselage the minimum principle stress is equal to theaxial stress and the maximum principle stress is equal to the transverse tension stress D.7
  • from pressurization. Figure 3.8 shows the minimum principle stress in the belly offuselage section 46 calculated by the MATLAB code. ⎛ 19 ft × 12 in ⎞ ⎜ ⎟ ( − − 2.725 × 10 lb ⋅ in ⎜ − 8 ) ft − (− 22.09in )⎟ − M zc ⎜ 2 ⎟σx = = ⎝ ⎠ = −31,875 psi 4 I 785,731inσ 1 = 18.74ksiσ 2 = −31.88ksi σ2Figure D.8: Minimum Principle Stress in Belly (psi)Maximum Shear StressThere is no shear fracture of shear buckling failure in fuselage section 46 but themaximum shear stress is calculated for verification of the MATLAB code. The shearflow equation is divided by two (i.e. Δq = V y Qz 2I ) since there are two shear paths for atube in pure bending. Also the very most top and bottom points of a tube have zeroin-plane shear stress for a tube in pure bending. Similar to the second moment of area D.8
  • the first moment of area needs to be uncoupled to account for a center of mass that isnot located at the axis-symmetric center of the tube. This is shown in the calculationbelow. Figure 3.9 shows the max in-plane shear stress calculated in the MATLABcode. The max shear stress is located in the crown skin so the crown skin thicknesstcrown _ skin , is used to calculate the maximum shear stress. πQ x = ∫ ydA 0Q z = 2r 2 (t crown ) + Acrown (0 − y g ) = ′ 2 ⎛ 19 ft ×12 in ⎞ ⎜ ft ⎟2⎜ ( 2 ) ⎟ (0.1120in ) + 43.69in [0 − (− 22.09 )] = 4,136in 3 ⎜ 2 ⎟ ⎝ ⎠q= V y Qz = (5.92 × 10 lbs )(4,136in ) = 1,558 lbs 5 3 2I z (2)(785,731in ) 4 in lbs 1,558 q in = 14,233 psiτ xy _ max_ crown = = t crown _ skin 0.1095in D.9
  • τ xy _ max r yFigure D.9: Max In-plane Shear Stress (psi)Buckling StabilityBuckling Strength of Belly StringersCripplingTo evaluate the buckling stability of a hat stiffener the cross section is broken up intoa series of flat plates as shown in Figure 3.10. The edges of the plates that do notdeflect during local buckling are subtracted as shown in Figure 3.10. New panel bwidths b1 , b2 , and 3 are calculated below to determine the crippling stress. D.10
  • b3 b2 b1Figure D.10: Idealized Stringer Sectionb1 = b f − t = 0.5in − 0.1106in = 0.3894inb2 = bw − 2t = 1.25in − 2(0.1106in) = 1.0288inb3 = bc − 2t = 1.5in − 2(0.1106in ) = 1.2788inThe crippling stress is calculated for each of the hat stiffener panels. The flanges areevaluated as one edge free(OEF), and the webs and caps as no edge free(NEF) Econdition. E x and y are the in-plane and transverse Modulus of Elasticityrespectively. D11 is the in-plane flexural stiffness. E is the Flexural Modulus ofElasticity. Fcc (= σ cc ) is the crippling stress in the military handbook notation.Fcu (= σ cu ) is the ultimate compression stress in military handbook notation. b is thewidth of the panel. t is the belly stringer thickness. The crippling stress of the hatsection is a weighted average of the crippling stress of the individual panels. Figure3.11 shows the crippling stress calculated by the MATLAB code. D.11
  • −0.786F cc Ex ⎛b E Fcu ⎞ (OEF ) = 0.5832 ⎜ ⎟F cu E ⎜ t Ex Ex E y ⎟ ⎝ ⎠ −0.842F cc Ex ⎛b E Fcu ⎞ ( NEF ) = 0.9356 ⎜ ⎟F cu E ⎜ t Ex Ex E y ⎟ ⎝ ⎠where 12(1 − ν xy 2 ) D11E= t3E x = E y = 8.6msiF cu = 70ksiD11 = 0.09523E x t stringer + 52.12 = 0.09523(8.6msi )(0.1106in ) + 52.12 = 1160.11lbs ⋅ in 3 3 12(1 − 0.33 2 )(1160.1lbs ⋅ in )E= = 9.17 msi (0.1106in )3F cc ⎞ − 0.784 ⎛ 70 × 10 3 psi ⎞ ⎛ 9.17 msi ⎞ ⎟ = 0.5832⎜ 0.3894in 9.17msi ⎟ ⎜ ⎟ = 1.45F cu ⎟ ⎜ 0.1106in 8.60msi 8.6 × 10 6 psi ⎟ ⎝ 8.60msi ⎠ ⎝ ⎠ ⎠ 1F cc ⎞ − 0.842 ⎛ 1.0288in 9.17 msi 70 × 10 3 psi ⎞ ⎛ 9.17 msi ⎞ ⎟ cu ⎟ = 0.9356⎜ ⎟ ⎜ 0.1106in 8.60msi 8.6 × 10 6 psi ⎟ ⎜ ⎟ = 1.095F ⎠ ⎝ ⎠ ⎝ 8.60msi ⎠ 2F cc ⎞ − 0.842 ⎛ 1.2788in 9.17 msi 70 × 10 3 psi ⎞ ⎛ 9.17 msi ⎞ ⎟ cu ⎟ = 0.9356⎜ ⎟ ⎜ 0.1106in 8.60msi 8.6 × 10 6 psi ⎟ ⎜ ⎟ = 0.912F ⎠ ⎝ ⎠ ⎝ 8.60msi ⎠ 3 ⎡ cc ⎢F ⎞⎤ ⎡ cc ⎤ ⎢F ⎞ ⎥ F cc ⎞ ⎟ ⎥b 1 + 2⎢ ⎟ b + ⎟b ⎟⎥ cu ⎟ ⎥ 2 cu ⎟ 2⎢ 3F ⎞ cc cu ⎟ = ⎢F ⎣ ⎠⎥ 1⎦ ⎢F ⎣ ⎠2⎥⎦ F ⎠ 3 cu ⎟ 2b1 + 2b2 + b3F ⎠ Hat _ Stiffener 2(1.450 )(0.3894in ) + 2(1.095)(1.0288in ) + (0.912 )(1.2788in )= = 1.105 2(0.3894in + 1.0288in ) + 1.2788inσ cc = Fcc = (1.105)(70ksi ) = 77.37ksi D.12
  • σ ccFigure D.11: Crippling Stress of Belly StringerLocal Center of MassFigure 3.12 shows the geometry of the hat stiffener. For this calculation y i is thelateral position of the hat stiffeners individual panels center of mass. θ is the angle ofthe hat stiffeners webs. The hat stiffener is symmetric about its lateral center so thewebs are both at the same angle. The hat stiffener center of mass is the weightedaverage of the individual panel’s center of mass. bc θ bw y yc yw yf bf *yi is lateral location of biFigure D.12: Geometry of Hat Stiffeneryf = 0y w = 0.5iny c = 1.0in 2(0 )(0.5in ) + 2(0.5in )(1.25in ) + (1.5in )(1.0in )y G _ hat = = 0.55in 2(0.5in + 1.25in ) + 1.5in D.13
  • Local Second Moment of AreaThe second moment of area is calculated below. It is calculated for each panel andthen added together. Each panels second moment of area is the sum of their localsecond moment of area and the parallel axis theorem calculation with respect to thecenter of mass of the cross section. As in previous calculations the subscripts; f , w ,and c stand for the flange, web, and cap respectively. ⎡1 ( ) 2⎤I f = 2 ⎢ b f t stringer + b f t stringer y f − y g _ hat ⎥ = 3 ⎣12 ⎦ ⎡1 2⎤2⎢ (0.5in )(0.1106in ) + (0.5in )(0.1106in )(0 − 0.55in ) ⎥ = 0.0336in 4 3 ⎣12 ⎦I w = 2 ⎢ bw t stringer (1 − cos(2θ ) ) + bw t stringer ( y w − y g _ hat ) ⎥ = ⎡1 3 2⎤ ⎣ 24 ⎦ ⎡1 ( ) 2⎤2⎢ (1.25in ) (0.1106in )(1 − cos 2 × 53 ) + (1.25in )(0.1106in )(0.5in − 0.55in ) ⎥ = 0.0237in 4 3 ⎣ 24 ⎦ 1I c = bc t stringer + bc t stringer ( y c − y g _ hat ) 2 = 3 12 1 (1.5in )(0.1106)3 + (1.5in )(0.1106in )(1.0in − 0.55in) 2 = 0.0338in 412I hat = I f + I w + I c = 0.0336in + 0.0237in + 0.0338in = 0.0911in 4Area [ ]Ahat = 2(b f + bw ) + bc t stringer = [2(0.5in + 1.25in ) + 1.5in](0.1106in ) = 0.553in 2Radius of Gyration I 0.0911in 4ρ= = = 0.4058in A 0.5530in 2 D.14
  • Effective LengthL = 20inc = 1.5 L 20inLe = = = 16.33in c 1.5Actual and Critical Slenderness RatioLe 16.33in = = 40.24 ρ 0.4058inLe ⎞ 2 2 ⎟ =π ⎟ =π = 46.84ρ ⎠ crit σ cc 77.37 × 10 psi 3 Ex 8.60 × 10 6 psiSince the critical slenderness ratio is larger than that of the stringer, the stringer willbuckle in-elastically.Stringer Buckling Strength ⎡ 1 σ ⎛L ⎞ 2 ⎤ ⎡ 1 77.37 × 10 3 psi ⎛ 16.33in ⎞ ⎤ 2σ cr = σ cc ⎢1 − 2 cc ⎜ e ⎜ ⎟ ⎟ ⎥ = 77.37 ksi ⎢1 − ⎜ ⎟ ⎥ ⎢ 4π E x ⎝ ρ ⎢ 4π 8.60 × 10 psi ⎝ 0.4058in ⎠ ⎥ 2 6 ⎣ ⎠ ⎥ ⎦ ⎣ ⎦= 48.81ksi D.15
  • Effective Width t skinThe belly skin thickness is used to calculate the effective width of the bellystringers. The safety factor S .F . = 1.5 is used because the skin buckles at limit load. E x _ skin (1.5)E x (1.5)8.60 × 10 6 psiwe = (0.85)t skin = (0.85)t skin = (0.85)(0.1858in ) 1.5 E x _ stringer σ stiff σ cr 49.05 × 10 3 psi= 2.52in y we weFigure D.13: Hat Stiffener Including Effective WidthLocal Center of Mass Including Effective Widthy we = 0 t skin 0.1825iny ′ _ hat = y g _ hat + g = 0.55in + = 0.6413in 2 2y′ = Ahat y ′ _ hat + 2 we t skin y we g = ( 0.5530in 2 (0.6413in ) + 0 ) = 0.2408in Ahat + 2 we t skin 0.5530in 2 + 2(2.520in )(0.1825in ) g D.16
  • Local Second Moment of Area Including Effective WidthThe subscript we , represent the second moment of area of the effective width of theskin.I we = 1 (2)we t skin 3 + 2we t skin ( y we − y ′g ) 2 12= 2 (2.520in )(0.1825in )3 + 2(2.520in )(0.1825in )(0 − 0.2408in) 2 = 0.0559in 4 12 ′ g g ( )I hat = I hat + Ahat ( y ′ _ hat − y ′ ) 2 = 0.0911in 4 + 0.5530in 2 (0.6413in − 0.2408in ) = 0.1798in 4 2I ′ = I hat + I we = 0.0559in 4 + 0.1798in 4 = 0.2357in 4 ′Area Including Effective WidthA′ = Ahat + 2 we t skin = 0.5530in 2 + 2(2.520in )(0.1825in ) = 1.4728in 2Radius of Gyration Including Effective Width I 0.2357in 4ρ′ = = = 0.40in A 1.4728in 2Buckling Strength Including Effective Width ⎡ 1 σ cc ⎛ Le ⎞ ⎤ 2 ⎡ 1 77.37 × 10 3 psi ⎛ 16.330in ⎞ ⎤ 2σ cr = σ cc ⎢1 − 2 ⎜ ⎟ ⎥ = 77.37ksi ⎢1 − ⎜ ⎟ ⎜ ⎟ ⎥= ⎢ 4π E x ⎝ ρ ⎠ ⎦ ⎢ 4π 8.60 × 10 psi ⎝ 0.400in ⎠ ⎥ 2 6 ⎣ ⎥ ⎣ ⎦= 47.98ksiσ crFigure D.14: Belly Stringer Buckling Strength D.17
  • Margins of SafetyCrown skin failure in OHT fracture at limit loadThe OHT strength is σ OHT = 57ksi . σ OHT 57.00ksiM .S .OHT = −1 = − 1 = 0.0076 ≈ 0 σ1 56.57ksiBelly skin failure in OHC fracture at limit loadThe OHC strength is σ OHC = 32 . ksi σ OHC 32.00ksiM .S .OHC = −1 = − 1 = 0.0038 ≈ 0 σ2 31.88ksiBelly stringer failure in buckling at ultimate load σ cr 47.98ksiM .S . Beam _ Buckling = −1 = − 1 = 0.0031 ≈ 0 (1.5)σ x (1.5)(31.88ksi )The method used for sample calculations is accurate enough to show the criticalmargins of safety are approximately zero. Therefore the MATLAB optimization hasbeen verified with the sample calculations. Figure 3.15 verifies that the crown andbelly skin are fracture critical, the crown stingers are not critical and are limited byminimum gauge, and the belly stringer are buckling failure critical. The fracturemargin of safety equations are given in appendix B. D.18
  • Thickness(in) Failure Mode 1)Fracture 2)Buckling Fuselage Section 46 M.S. > 0, so crown stringers are not critical and are governed by minimum gauge. Margins of Safety σ1Figure D.15: Output for WB CFRP Baseline D.19
  • 2 ⎛t⎞ t3σ cr = K c E ⎜ ⎟ → N x _ cr = K c E 2 ⎝b⎠ b 2 ⎛t⎞ t3τ cr = K s E ⎜ ⎟ → N x _ cr = K s E 2 ⎝b⎠ b 1 ⎛ N w ⎞3tskin =⎜ x ⎟ ⎝ Kc E ⎠ 1 ⎛ N xy h ⎞3tskin =⎜ ⎟ ⎜KE ⎟ ⎝ s ⎠FuselageAxial load from bending onlyUn_pressurized t 2tσ cr = Cb E = Cb E r D 2t 2N cr = Cb E D Nx Dt= 2Cb E D.20
  • Sensitivity Criteria σAM .S . = −1 = 0 → σ A = σ D σD NxσD = t∂σ D N = − 2x ∂t t σ 2 ⎛t⎞σ A = KE ⎜ ⎟ = CEt 2 → E = A2 b ⎝ ⎠ Ct∂E σ N 2 ⎛ ∂σ ⎞ = −2 A3 = −2 x4 = 2 ⎜ D ⎟∂t Ct Ct Ct ⎝ ∂t ⎠ KC= 2 b∂E 2b 2 ⎛ ∂σ D ⎞ = ⎜ ⎟∂t Kt 2 ⎝ ∂t ⎠∂E ∂t ΔE 2 ⎛ b ⎞ 2 = = ⎜ ⎟∂σ ∂t Δσ K ⎝ t ⎠Un-Stiffened Configuration WeightsWB CFRP W.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) Nxy LC1 (lb/in) ttop_skin tbottom_skin tweb Weight (lb) 0% 20.3 8.12 3.8 2.80 2.07 0.72 6262 11% 20.7 8.28 3.8 2.65 1.95 0.65 4804 21% 20.9 8.36 3.9 2.49 1.84 0.59 5976 37% 20.6 8.24 4.2 2.22 1.63 0.48 1726 43% 19.9 7.96 4.1 2.10 1.55 0.47 3726 58% 16.2 6.48 3.8 1.71 1.26 0.36 2216 74% 9.4 3.76 2.7 1.18 0.87 0.27 967 90% 1.9 0.76 1.0 0.54 0.40 0.15 329 Total Weight (lb) 52,011 D.21
  • WB AluminumW.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) Nxy LC1 (lb/in) ttop_skin tbottom_skin tweb Weight (lb) 0% 19.46 7.79 3.8 3.06 2.25 0.59 12034 11% 19.97 7.99 3.9 2.89 2.13 0.53 9265 21% 20.28 8.11 3.9 2.73 2.01 0.48 11560 37% 20.05 8.02 4.2 2.43 1.79 0.40 3345 43% 19.43 7.77 4.1 2.30 1.70 0.38 7238 58% 15.92 6.37 3.8 1.88 1.38 0.30 4314 74% 9.26 3.71 2.7 1.30 0.96 0.22 1870 90% 1.76 0.71 1.0 0.58 0.43 0.12 628 Total Weight (lb) 100,509NB CFRPW.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) Nxy LC1 (lb/in) ttop_skin tbottom_skin tweb Weight (lb) 0% 8.32 3.33 1.5 1.57 1.15 0.40 1498 11% 8.48 3.39 1.6 1.48 1.09 0.36 1154 21% 8.57 3.43 1.6 1.40 1.03 0.33 1441 37% 8.44 3.38 1.7 1.24 0.92 0.28 417 43% 8.16 3.26 1.7 1.18 0.87 0.26 897 58% 6.64 2.66 1.5 0.95 0.70 0.21 536 74% 3.85 1.54 1.1 0.66 0.49 0.15 235 90% 0.78 0.31 0.4 0.30 0.22 0.09 80 Total Weight (lb) 12,515NB AluminumW.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) Nxy LC1 (lb/in) ttop_skin tbottom_skin tweb Weight (lb) 0% 7.98 3.19 1.6 1.71 1.26 0.33 2879 11% 8.19 3.27 1.6 1.62 1.19 0.30 2226 21% 8.31 3.32 1.6 1.53 1.13 0.27 2786 37% 8.22 3.29 1.7 1.36 1.01 0.23 808 43% 7.96 3.19 1.7 1.29 0.95 0.21 1742 58% 6.52 2.61 1.5 1.05 0.77 0.17 1042 74% 3.80 1.52 1.1 0.73 0.54 0.12 454 90% 0.72 0.29 0.4 0.33 0.24 0.07 152 Total Weight (lb) 24,178 D.22
  • WB CFRP F.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) (r/t)est Cb tavg Weight (lb) 25% 741 -2694 268 0.18 0.426 4607 37% 1641 -3787 226 0.18 0.505 5138 52% 3323 1177 248 0.19 0.461 6479 70% 2059 6615 184 0.21 0.618 6856 83% 791 635 466 0.16 0.245 2251 100% 320 258 685 0.14 0.166 1279 Total Weight (lb) 26,610WB Al F.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) (r/t)est Cb tavg Weight (lb) 25% 741 -2694 285 0.18 0.399 7711 37% 1641 -3787 254 0.20 0.449 8360 52% 3323 1177 264 0.19 0.432 10553 70% 2059 6615 192 0.20 0.594 11678 83% 791 635 497 0.16 0.230 3768 100% 320 258 731 0.14 0.156 2140 Total Weight (lb) 44,210NB CFRP F.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) (r/t)est Cb tavg Weight (lb) 25% 432 -1675 245 0.20 0.319 1311 37% 957 -2312 208 0.20 0.374 1455 52% 1938 687 228 0.20 0.343 1831 70% 1201 3858 161 0.20 0.484 2011 83% 461 370 430 0.17 0.181 625 100% 186 150 657 0.16 0.119 347 Total Weight (lb) 7,580NB Al D.23
  • F.S. (%) Nx LC1 (lb/in) Nx LC2 (lb/in) (r/t)est Cb tavg Weight (lb) 25% 432 -1675 252 0.19 0.309 2273 37% 957 -2312 220 0.20 0.354 2488 52% 1938 687 240 0.20 0.324 3094 70% 1201 3858 175 0.21 0.447 3338 83% 461 370 455 0.17 0.172 1070 100% 186 150 672 0.15 0.116 606 Total Weight (lb) 12,868 D.24